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  COLLECTION OF FORMULAE..........for those who did not see this..........!!!!!!!!!!!!!!!   15 Nickels awarded!
Tagged with:  academic    [Post New]posted on 14 Aug 2007 14:30:25 IST    

i hav not posted it bcas i want some rates........i hav already got 213 points for this article.....!!!!!!!!!This is for those who havent seen it yet.....!!!!!!!!!!!!

DEFINITE INTEGRALS THAT CONTAIN TRIGONOMETRIC FUNCTIONS

Note that all the constant are positive.
1.
$displaystyleint_{0}^{pi}sin mx sin nx dx=left{ egin{array}{ll} displa... ...in} &mbox{if $m$space and $n$space integers and};; m=n end{array} ight. $
2.
$displaystyleint_{0}^{pi} cos mx cos nx dx=left{ egin{array}{ll} displ... ...in} &mbox{if $m$space and $n$space integers and};; m=n end{array} ight. $
3.
$displaystyleint_{0}^{pi}sin mx cos nx dx=left{ egin{array}{ll} displa... ... $m$space and $n$space integers and $m+n$space even} end{array} ight. $
4.
$displaystyleint_{0}^{pi/2} sin^2 x dx=int_{0}^{pi/2}cos^2 dx=displaystyle rac{pi}{4}$
5.
$displaystyleint_{0}^{pi/2}sin^{2m} x dx=int_{0}^{pi/2}cos^{2m} x dx=dis... ...cdot 4cdot 6cdot cdotcdotcdot 2m}left(displaystyle rac{pi}{2} ight)$,
m=1,2,...
6.
$displaystyleint_{0}^{pi/2}sin^{2m+1}x dx=int_{0}^{pi/2}cos^{2m+1}x dx=d... ...c{2cdot 4cdot 6cdotcdotcdot 2m}{1cdot 3cdot 5cdot cdotcdotcdot 2m+1}$,
m=1,2,...
7.
$displaystyleint_{0}^{pi/2}sin^{2p-1}x cos^{2q-1}x dx=displaystyle rac{Gamma(p)Gamma(q)}{2Gamma(p+q)}$
8.
$displaystyleint_{0}^{infty}displaystyle rac{sin px}{x}dx=left{ egin{... ...n} p>0 0&hspace{.3in} p=0 -pi/2&hspace{.3in} p<0 end{array} ight. $
9.
$displaystyleint_{0}^{infty}displaystyle rac{sin pxcos qx}{x}dx=left{ ... ...0 pi/2&hspace{.3in} 0<p<q pi/4&hspace{.3in} p=q>0 end{array} ight. $
10.
$displaystyleint_{0}^{infty}displaystyle rac{sin px sin qx}{x^2}dx=left... ...hspace{.3in} 0<pleq q pi q/2&hspace{.3in} pgeq q>0 end{array} ight. $
11.
$displaystyleint_{0}^{infty}displaystyle rac{sin^2 px}{x^2}dx=displaystyle rac{pi p}{2}$
12.
$displaystyleint_{0}^{infty}displaystyle rac{1-cos px}{x^2}dx=displaystyle rac{pi p}{2}$
13.
$displaystyleint_{0}^{infty}displaystyle rac{cos px-cos qx}{x}dx=lndisplaystyle rac{q}{p}$
14.
$displaystyleint_{0}^{infty}displaystyle rac{cos px-cos qx}{x^2}dx=displaystyle rac{pi(q-p)}{2}$
15.
$displaystyleint_{0}^{infty}displaystyle rac{cos mx}{x^2+a^2}dx=displaystyle rac{pi}{2a}e^{-ma}$
16.
$displaystyleint_{0}^{infty}displaystyle rac{xsin mx}{x^2+a^2}dx=displaystyle rac{pi}{2}e^{-ma}$
17.
$displaystyleint_{0}^{infty}displaystyle rac{sin mx}{x(x^2+a^2)}dx=displaystyle rac{pi}{2a^2}(1-e^{-ma})$
18.
$displaystyleint_{0}^{2pi}displaystyle rac{dx}{a+bsin x}=displaystyle rac{2pi}{displaystyle sqrt{a^2-b^2}}$
19.
$displaystyleint_{0}^{2pi}displaystyle rac{dx}{a+bcos x}=displaystyle rac{2pi}{displaystyle sqrt{a^2-b^2}}$
20.
$displaystyleint_{0}^{pi/2}displaystyle rac{dx}{a+bcos x}=displaystyle rac{cos^{-1}(b/a)}{displaystyle sqrt{a^2-b^2}}$
21.
$displaystyleint_{0}^{2pi}displaystyle rac{dx}{(a+bsin x)^2}=int_{0}^{2... ...playstyle rac{dx}{(a+bcos x)^2}=displaystyle rac{2pi a}{(a^2-b^2)^{3/2}}$
22.
$displaystyleint_{0}^{2pi}displaystyle rac{dx}{1-2acos x+a^2}=displaystyle rac{2pi}{1-a^2},hspace{.2in}0<a<1$
23.
$displaystyleint_{0}^{pi}displaystyle rac{x sin x dx}{1-2acos x +a^2}=l... ...mid amid <1 piln(1+1/a) &hspace{.3in} mid amid >1 end{array} ight. $
24.
$displaystyleint_{0}^{pi}displaystyle rac{cos mx dx}{1-2acos x+a^2}=displaystyle rac{pi a^m}{1-a^2},hspace{.2in}a^2<1$,
m=0,1,2,...
25.
$displaystyleint_{0}^{infty}sin ax^2 dx=int_{0}^{infty}cos ax^2 dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{2a}}$
26.
$displaystyleint_{0}^{infty}sin ax^n dx=displaystyle rac{1}{na^{1/n}}Gamma(1/n)sindisplaystyle rac{pi}{2n},hspace{.2in}n>1$
27.
$displaystyleint_{0}^{infty}cos ax^n dx=displaystyle rac{1}{na^{1/n}}Gamma(1/n)cosdisplaystyle rac{pi}{2n},hspace{.2in}n>1$
28.
$displaystyleint_{0}^{infty}displaystyle rac{sin x}{displaystyle sqrt{x... ...s x}{displaystyle sqrt{x}}dx=displaystyle sqrt{displaystyle rac{pi}{2}}$
29.
$displaystyleint_{0}^{infty}displaystyle rac{sin x}{x^p}dx=displaystyle rac{pi}{2Gamma (p)sin(ppi/2)},hspace{.2in}0<p<1$
30.
$displaystyleint_{0}^{infty}displaystyle rac{cos x}{x^p}dx=displaystyle rac{pi}{2Gamma (p)cos(ppi/2)},hspace{.2in}0<p<1$
31.
$displaystyleint_{0}^{infty}sin ax^2 cos 2bx dx=displaystyle rac{1}{2}d... ...}}left( cosdisplaystyle rac{b^2}{a}-sindisplaystyle rac{b^2}{a} ight)$
32.
$displaystyleint_{0}^{infty}cos ax^2cos 2bxdx=displaystyle rac{1}{2}dis... ...}}left( cosdisplaystyle rac{b^2}{a}+sindisplaystyle rac{b^2}{a} ight)$
33.
$displaystyleint_{0}^{infty}displaystyle rac{sin^3 x}{x^3}dx=displaystyle rac{3pi}{8}$
34.
$displaystyleint_{0}^{infty}displaystyle rac{sin^4 x}{x^4}dx=displaystyle rac{pi}{3}$
35.
$displaystyleint_{0}^{infty}displaystyle rac{ an x}{x}dx=displaystyle rac{pi}{2}$
36.
$displaystyleint_{0}^{pi/2}displaystyle rac{dx}{1+ an^m x}=displaystyle rac{pi}{4}$
37.
$displaystyleint_{0}^{pi/2}displaystyle rac{x}{sin x}dx=2left{ display... ...displaystyle rac{1}{5^2}-displaystyle rac{1}{7^2}+cdotcdotcdot ight}$
38.
$displaystyleint_{0}^{1}displaystyle rac{ an^{-1}x}{x}dx=displaystyle r... ...}{3^2}+displaystyle rac{1}{5^2}-displaystyle rac{1}{7^2}+ cdotcdotcdot$
39.
$displaystyleint_{0}^{1}displaystyle rac{sin^{-1}x}{x}dx=displaystyle rac{pi}{2}ln2$


DEFINITE INTEGRALS CONTAINING HYPERBOLIC FUNCTIONS

1.
$displaystyleint_{0}^{infty}displaystyle rac{sin ax}{sinh bx}dx=displaystyle rac{pi}{2b} anhdisplaystyle rac{api}{2b}$
2.
$displaystyleint_{0}^{infty}displaystyle rac{cos ax}{cosh bx}dx=displaystyle rac{pi}{2b}displaystyle rac{1}{cosh (api/2b)}$
3.
$displaystyleint_{0}^{infty}displaystyle rac{x dx}{sinh ax}=displaystyle rac{pi^2}{4a^2}$
4.
$displaystyleint_{0}^{infty}displaystyle rac{x^n dx}{sinh ax}=displaysty... ...tyle rac{1}{2^{n+1}}+displaystyle rac{1}{3^{n+1}}+cdotcdotcdot ight}$
5.
$displaystyleint_{0}^{infty}displaystyle rac{sinh ax}{e^{bx}+1}dx=displaystyle rac{pi}{2b}cscdisplaystyle rac{api}{b}-displaystyle rac{1}{2a}$
6.
$displaystyleint_{0}^{infty}displaystyle rac{sinh ax}{e^{bx}-1}dx=displaystyle rac{1}{2a}-displaystyle rac{pi}{2b}cotdisplaystyle rac{api}{b}$



First and Second Order Differential Equations

First Order Differential equations

A first order differential equation is of the form:
displaymath137

Linear Equations:

displaymath139
The general general solution is given by
displaymath141
where
displaymath143
is called the integrating factor.

Separable Equations:

displaymath145
(1)
Solve the equation g(y) = 0 which gives the constant solutions.
(2)
The non-constant solutions are given by
displaymath149

Bernoulli Equations:

displaymath151
(1)
Consider the new function tex2html_wrap_inline153 .
(2)
The new equation satisfied by v is
displaymath157
(3)
Solve the new linear equation to find v.
(4)
Back to the old function y through the substitution tex2html_wrap_inline163 .
(5)
If n > 1, add the solution y=0 to the ones you got in (4).

Homogenous Equations:

displaymath137
is homogeneous if the function f(x,y) is homogeneous, that is
displaymath173
By substitution, we consider the new function
displaymath175
The new differential equation satisfied by z is
displaymath179
which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by
displaymath183
Do not forget to go back to the old function y = xz.

Exact Equations:

displaymath187
is exact if
displaymath189
The condition of exactness insures the existence of a function F(x,y) such that
displaymath193
All the solutions are given by the implicit equation
displaymath195

Second Order Differential equations


Homogeneous Linear Equations with constant coefficients:

displaymath197
Write down the characteristic equation
displaymath199
(1)
If tex2html_wrap_inline201 and tex2html_wrap_inline203 are distinct real numbers (this happens if tex2html_wrap_inline205 ), then the general solution is
displaymath207
(2)
If tex2html_wrap_inline209 (which happens if tex2html_wrap_inline211 ), then the general solution is
displaymath213
(3)
If tex2html_wrap_inline201 and tex2html_wrap_inline203 are complex numbers (which happens if tex2html_wrap_inline219 ), then the general solution is
displaymath221
where
displaymath223
that is
displaymath225

Non Homogeneous Linear Equations:

displaymath227
The general solution is given by
displaymath229
where tex2html_wrap_inline231 is a particular solution and tex2html_wrap_inline233 is the general solution of the associated homogeneous equation
displaymath235
In order to find tex2html_wrap_inline237 two major techniques were developed.

Method of undetermined coefficients or Guessing Method

This method works for the equation
displaymath239
where a, b, and c are constant and
displaymath247
where tex2html_wrap_inline249 is a polynomial function with degree n. In this case, we have
displaymath253
where
displaymath255
The constants tex2html_wrap_inline257 and tex2html_wrap_inline259 have to be determined. The power s is equal to 0 if tex2html_wrap_inline265 is not a root of the characteristic equation. If tex2html_wrap_inline265 is a simple root, then s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x) satisfies the following
displaymath275
where tex2html_wrap_inline277 are of the forms cited above, then we split the original equation into N equations
displaymath281
then find a particular solution tex2html_wrap_inline283 . A particular solution to the original equation is given by
displaymath285

Method of Variation of Parameters

This method works as long as we know two linearly independent solutions tex2html_wrap_inline287 of the homogeneous equation
displaymath289
Note that this method works regardless if the coefficients are constant or not. a particular solution as
displaymath291
where tex2html_wrap_inline293 and tex2html_wrap_inline295 are functions. From this, the method got its name.
The functions tex2html_wrap_inline293 and tex2html_wrap_inline295 are solutions to the system:
displaymath301
which implies
displaymath303
Therefore, we have
displaymath305

Euler-Cauchy Equations:

displaymath307
where b and c are constant numbers. By substitution, set
displaymath313
then the new equation satisfied by y(t) is
displaymath317
which is a second order differential equation with constant coefficients.
(1)
Write down the characteristic equation
displaymath129
(2)
If the roots tex2html_wrap_inline201 and tex2html_wrap_inline203 are distinct real numbers, then the general solution is given by
displaymath130
(2)
If the roots tex2html_wrap_inline201 and tex2html_wrap_inline203 are equal ( tex2html_wrap_inline209 ), then the general solution is
displaymath131
(3)
If the roots tex2html_wrap_inline201 and tex2html_wrap_inline203 are complex numbers, then the general solution is
displaymath132
where tex2html_wrap_inline339 and tex2html_wrap_inline341 .



Common Integrals

1.
$displaystyle int adx=ax$
2.
$displaystyle int af(x)dx=a displaystyle int f(x)dx$
3.
$displaystyle int left( u pm v pm w pm cdots ight) dx = displaystyle int udx pm displaystyle int vdx pm displaystyle int wdx pm cdots $
4.
$displaystyle int udv = uv - displaystyle int vdu$
5.
$displaystyle int f(ax)dx = displaystyle rac{1}{a} displaystyle int f(u)du$
6.
$displaystyle int F{f(x)}dx = displaystyle int F(u) displaystyle rac{dx}{du}du = displaystyle int displaystyle rac{F(u)}{f'(x)}du$
7.
$displaystyle int u^{n}du = displaystyle rac{u^{n+1}}{n+1}, n eq -1$
8.
$egin{array}{lcl} displaystyle intdisplaystyle rac{du}{u} & = & ln u mb... ...or} ln (-u) mbox{ if} u<0 & = & ln left ert u ight ert end{array}$
9.
$displaystyle int e^{u}du=e^{u}$
10.
$displaystyle int a^{u}du = int e^{u ln a}du = displaystyle rac{e^{u ln a}}{ln a} = displaystyle rac{a^{u}}{ln a} , a >0, a eq 1$
11.
$displaystyle int sin u du = -cos u$
12.
$displaystyle int cos u du = sin u$
13.
$displaystyle int an u du = ln sec u = -ln cos u$
14.
$displaystyle int cot u du = ln sin u$
15.
$displaystyle int sec u du = ln (sec u + an u) = ln an left( displaystyle rac{u}{2} + displaystyle rac{pi}{4} ight)$
16.
$displaystyle int csc u du = ln (csc u - cot u) = ln an displaystyle rac{u}{2}$

17.
$displaystyle int sec ^{2} u du = an u$
18.
$displaystyle int csc ^{2} u du = -cot u$
19.
$displaystyle int an ^{2} u du = an u - u$
20.
$displaystyle int cot ^{2} u du = -cot u - u $
21.
$displaystyle int sin ^{2} u du = displaystyle rac{u}{2} - displaystyle rac{sin 2u}{4} = displaystyle rac{1}{2} (u-sin u cos u)$
22.
$displaystyle int cos ^{2} u du = displaystyle rac{u}{2} + displaystyle rac{sin 2u}{4} = displaystyle rac{1}{2} (u+sin u cos u)$
23.
$displaystyle int sec u an u du = sec u$
24.
$displaystyle int csc u cot u du = -csc u $
25.
$displaystyle int sinh u du = cosh u$
26.
$displaystyle int cosh u du = sinh u$

27.
$displaystyle int anh u du = ln cosh u$
28.
$displaystyle int coth u du = ln sinh u$
29.
$displaystyle int $sech $u du = sin ^{-1}( anh u )$ or $2 an ^{-1}e^{u}$
30.
$displaystyle int $csch $ u du = ln anh displaystyle rac{u}{2}$ or $-coth ^{-1}e^{u}$
31.
$displaystyle int $sech $^{2} u du = anh u $
32.
$displaystyle int $csch 2 u du =-coth u
33.
$displaystyle int anh ^{2} u du = u - anh u$
34.
$displaystyle int $coth 2 u du = u -coth u
35.
$displaystyle intsinh ^{2} u du = displaystyle rac{sinh 2u}{4} - displaystyle rac{u}{2} = displaystyle rac{1}{2}(sinh u cosh u- u)$
36.
$displaystyle intcosh ^{2} u du = displaystyle rac{sinh 2u}{4} + displaystyle rac{u}{2} = displaystyle rac{1}{2}(sinh u cosh u+ u)$

37.
$displaystyle int $sech $ u anh u du = - $sech u
38.
$displaystyle int $csch ucoth u du = -csch u
39.
$displaystyle intdisplaystyle rac{du}{u^{2}+a^{2}} = displaystyle rac{1}{a} an^{-1} displaystyle rac{u}{a}$
40.
$displaystyle intdisplaystyle rac{du}{u^{2} - a^{2}}= displaystyle rac{1... ...n left( displaystyle rac{u - a}{u+a} ight) = - displaystyle rac{1}{a} $coth $ ^{-1} displaystyle rac{u}{a} , u^{2}>a^{2}$
41.
$displaystyle intdisplaystyle rac{du}{a^{2}-u^{2}}= displaystyle rac{1}{... ...= displaystyle rac{1}{a} anh ^{-1} displaystyle rac{u}{a} , u^{2}<a^{2}$
42.
$displaystyle intdisplaystyle rac{du}{sqrt{a^{2}-u^{2}}} = sin ^{-1} displaystyle rac{u}{a}$
43.
$displaystyle intdisplaystyle rac{du}{sqrt{u^{2}+a^{2}}} = ln left( u+ displaystylesqrt{u^{2} + a^{2}} ight)$ or $ sinh ^{-1} displaystyle rac{u}{a}$
44.
$displaystyle intdisplaystyle rac{du}{sqrt{u^{2}-a^{2}}} = ln left( u + displaystylesqrt{u^{2} - a^{2}} ight)$
45.
$displaystyle intdisplaystyle rac{du}{u sqrt{u^{2}-a^{2}}} = displaystyle rac{1}{a} sec ^{-1} left ert displaystyle rac{u}{a} ight ert$
46.
$displaystyle intdisplaystyle rac{du}{u sqrt{u^{2}+a^{2}}}=-displaystyle rac{1}{a} ln left( displaystyle rac{a+sqrt{u^{2}+a^{2}}}{u} ight)$
47.
$displaystyle intdisplaystyle rac{du}{u sqrt{a^{2}-u^{2}}}=-displaystyle rac{1}{a} ln left( displaystyle rac{a+sqrt{a^{2}-u^{2}}}{u} ight)$
48.
$displaystyle int f^{(n)}gdx = f^{(n-1)}g - f^{(n-2)}g' + f^{(n-3)} g'' - cdots (-1)^{n} displaystyle int fg^{(n)}dx$



COMMON SUBSTITUTIONS

1.
$displaystyle int F(ax+b)dx = displaystyle rac{1}{a} displaystyle int F(u)du$
where $u=ax,+,b$
2.
$displaystyle int Fleft(displaystylesqrt{ax, +, b} ight), dx = displaystyle rac{2}{a} displaystyle int u,F(u),du $
where $u=displaystylesqrt{ax,+,b}$
3.
$displaystyle int Fleft( sqrt[n]{ax+b} ight) ,dx = displaystyle rac{n}{a} displaystyle int u^{n-1},F(u),du$
where $u=sqrt[n]{ax+b}$
4.
$displaystyle int Fleft( displaystylesqrt{a^{2}-x^{2}} ight),dx = a,displaystyle int F(a cos u),cos u,du$
where $x=asin u$
5.
$displaystyle int Fleft( displaystylesqrt{x^2+a^{2}} ight),dx= a,displaystyle int F(a sec u) sec ^{2} u , du$
where $x=a an u$
6.
$displaystyle int Fleft( displaystylesqrt{x^{2}-a^{2}} ight),dx=a displaystyle int F(a an u) sec u an u,du$
where $x=asec u$
7.
$displaystyle int F (edisplaystyle^{ax}),dx = displaystyle rac{1}{a} displaystyle intdisplaystyle rac{F(u)}{u},du$
where $u=edisplaystyle^{ax}$
8.
$displaystyle int F(ln x),dx = displaystyle int F(u),edisplaystyle^u,du$
where $u=ln x$
9.
$displaystyle int Fleft( sin ^{-1}displaystyle rac{x}{a} ight),dx = a,displaystyle int F(u)cos u,du$
where $u=sin ^{-1}displaystyle rac{x}{a}$

DEFINITE INTEGRALS CONTAINING EXPONENTIAL FUNCTIONS

1.
$displaystyleint_{0}^{infty}e^{-ax}cos bx dx=displaystyle rac{a}{a^2+b^2}$
2.
$displaystyleint_{0}^{infty}e^{-ax}sin bx dx=displaystyle rac{b}{a^2+b^2}$
3.
$displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}sin bx}{x}dx= an^{-1}displaystyle rac{b}{a}$
4.
$displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}-e^{-bx}}{x}dx=lndisplaystyle rac{b}{a}$
5.
$displaystyleint_{0}^{infty}e^{-ax^2}dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}$
6.
$displaystyleint_{0}^{infty}e^{-ax^2}cos bx dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}e^{-b^2/4a}$
7.
$displaystyleint_{0}^{infty}e^{-(ax^2+bx+c)}dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}e^{(b^2-4ac)/4a}$
8.
$displaystyleint_{-infty}^{infty}e^{-(ax^2+bx+c)}dx=displaystyle sqrt{displaystyle rac{pi}{a}}e^{(b^2-4ac)/4a}$
9.
$displaystyleint_{0}^{infty}x^n e^{-ax}dx=displaystyle rac{Gamma(n+1)}{a^n+1}$
10.
$displaystyleint_{0}^{infty}x^m e^{-ax^2}dx=displaystyle rac{Gamma[(m+1)/2]}{2a^{(m+1)/2}}$
11.
$displaystyleint_{0}^{infty}e^{-(ax^2+b/x^2)}dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}e^{-2displaystyle sqrt{ab}}$
12.
$displaystyleint_{0}^{infty}displaystyle rac{xdx}{e^x-1}=displaystyle r... ...3^2}+displaystyle rac{1}{4^2}+cdotcdotcdot =displaystyle rac{pi^2}{6}$
13.
$displaystyleint_{0}^{infty}displaystyle rac{x^{n-1}}{e^x-1}dx=Gamma(n+1)... ...displaystyle rac{1}{2^n}+displaystyle rac{1}{3^n}+cdotcdotcdot ight)$
14.
$displaystyleint_{0}^{infty}displaystyle rac{xdx}{e^x+1}=displaystyle r... ...2}-displaystyle rac{1}{4^2}+ cdotcdotcdot =displaystyle rac{pi^2}{12}$
15.
$displaystyleint_{0}^{infty}displaystyle rac{x^{n-1}}{e^x+1}dx=Gamma(n+1)... ...displaystyle rac{1}{2^n}+displaystyle rac{1}{3^n}-cdotcdotcdot ight)$
16.
$displaystyleint_{0}^{infty}displaystyle rac{sin mx}{e^{2pi x}-1}dx=displaystyle rac{1}{4}cothdisplaystyle rac{m}{2}-displaystyle rac{1}{2m}$
17.
$displaystyleint_{0}^{infty}left( displaystyle rac{1}{1+x}-e^{-x} ight)displaystyle rac{dx}{x}=gamma$
where the constant $gamma$ is the eulers constant.
18.
$displaystyleint_{0}^{infty}displaystyle rac{e^{-x^2}-e^{-x}}{x}dx=displaystyle rac{1}{2}gamma$
where the constant $gamma$ is the EULERs CONSTANT.
19.
$displaystyleint_{0}^{infty}left( displaystyle rac{1}{e^x-1}-displaystyle rac{e^{-x}}{x} ight)dx=gamma$
where the constant $gamma$ is the EULERs CONSTANT.
20.
$displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}-e^{-bx}}{xsec px}dx=displaystyle rac{1}{2}lnleft(displaystyle rac{b^2+p^2}{a^2+p^2} ight)$
21.
$displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}-e^{-bx}}{xcsc px}dx= an^{-1}displaystyle rac{b}{p}- an^{-1}displaystyle rac{a}{p}$
22.
$displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}(1-cos x)}{x^2}dx=cot^{-1}a-displaystyle rac{a}{2}ln(a^2+1)$


Integrals with Inverse Trigonometric Functions

1.
$displaystyleintsin^{-1}displaystyle rac{x}{a}dx=xsin^{-1} displaystyle rac{x}{a}+displaystyle sqrt{a^2-x^2}$
2.
$displaystyleint xsin^{-1}displaystyle rac{x}{a}dx=left(displaystyle r... ...displaystyle rac{x}{a}+displaystyle rac{xdisplaystyle sqrt{a^2-x^2}}{4}$
3.
$displaystyleint x^2sin^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...yle rac{x}{a}+displaystyle rac{(x^2+2a^2)displaystyle sqrt{a^2-x^2)}}{9}$
4.
$displaystyleintdisplaystyle rac{sin^{-1}(x/a)}{x}dx=displaystyle rac{x... ...rac{1cdot 3cdot 5(x/a)^7}{2cdot 4cdot 6cdot 7cdot 7} + cdot cdot cdot $
5.
$displaystyleintdisplaystyle rac{sin^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...rac{1}{a}lnleft(displaystyle rac{a+displaystyle sqrt{a^2-x^2}}{x} ight)$
6.
$displaystyleintleft(sin^{-1}displaystyle rac{x}{a} ight)^2 dx=xleft(s... ...a} ight)^2 -2x+2displaystyle sqrt{a^2-x^2}sin^{-1}displaystyle rac{x}{a}$
7.
$displaystyleintcos^{-1}displaystyle rac{x}{a}dx=xcos^{-1}displaystyle rac{x}{a}-displaystyle sqrt{a^2-x^2}$
8.
$displaystyleint xcos^{-1}displaystyle rac{x}{a}dx=left(displaystyle r... ...displaystyle rac{x}{a}-displaystyle rac{xdisplaystyle sqrt{a^2-x^2}}{4}$

9.
$displaystyleint x^2cos^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...tyle rac{x}{a}-displaystyle rac{(x^2+2a^2)displaystyle sqrt{a^2-x^2}}{9}$
10.
$displaystyleintdisplaystyle rac{cos^{-1}(x/a)}{x}dx=displaystyle rac{pi}{2}ln x-intdisplaystyle rac{sin^{-1}(x/a)}{x}dx$
11.
$displaystyleintdisplaystyle rac{cos^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...rac{1}{a}lnleft(displaystyle rac{a+displaystyle sqrt{a^2-x^2}}{x} ight)$
12.
$displaystyleintleft(cos^{-1}displaystyle rac{x}{a} ight)^2 dx=xleft(c... ...{a} ight)^2-2x-2displaystyle sqrt{a^2-x^2}cos^{-1}displaystyle rac{x}{a}$
13.
$displaystyleint an^{-1}displaystyle rac{x}{a}dx=x an^{-1}displaystyle rac{x}{a}-displaystyle rac{a}{2}ln(x^2+a^2)$
14.
$displaystyleint x an^{-1}displaystyle rac{x}{a}dx=displaystyle rac{1}{2}(x^2+a^2) an^{-1}displaystyle rac{x}{a}-displaystyle rac{ax}{2}$
15.
$displaystyleint x^2 an^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...frac{x}{a}-displaystyle rac{ax^2}{6}+displaystyle rac{a^3}{6}ln(x^2+a^2)$

16.
$displaystyleintdisplaystyle rac{ an^{-1}(x/a)}{x}dx=displaystyle rac{x... ...playstyle rac{(x/a)^5}{5^2}-displaystyle rac{(x/a)^7}{7^2}+cdotcdotcdot$
17.
$displaystyleintdisplaystyle rac{ an^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...{a}-displaystyle rac{1}{2a}lnleft(displaystyle rac{x^2+a^2}{x^2} ight)$
18.
$displaystyleintcot^{-1}displaystyle rac{x}{a}dx=xcot^{-1}displaystyle rac{x}{a}+displaystyle rac{a}{2}ln(x^2+a^2)$
19.
$displaystyleint xcot^{-1}displaystyle rac{x}{a}dx=displaystyle rac{1}{2}(x^2+a^2)cot^{-1}displaystyle rac{x}{a}+displaystyle rac{ax}{2}$
20.
$displaystyleint x^2cot^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...frac{x}{a}+displaystyle rac{ax^2}{6}-displaystyle rac{a^3}{6}ln(x^2+a^2)$
21.
$displaystyleintdisplaystyle rac{cot^{-1}(x/a)}{x}dx=displaystyle rac{pi}{2}ln x-intdisplaystyle rac{ an^{-1}(x/a)}{x}dx$

22.
$displaystyleintdisplaystyle rac{cot^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...{x}+displaystyle rac{1}{2a}lnleft(displaystyle rac{x^2+a^2}{x^2} ight)$
23.
$ displaystyleintsec^{-1}displaystyle rac{x}{a}dx=left{ egin{array}{ll... ...style rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight. $
24.
$displaystyleint xsec^{-1}displaystyle rac{x}{a}dx=left{ egin{array}{l... ...ystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight. $
25.
$displaystyleint x^2sec^{-1}displaystyle rac{x}{a}dx=left{ egin{array}... ...ystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight. $
26.
$displaystyleintdisplaystyle rac{sec^{-1}(x/a)}{x}dx=displaystyle rac{... ... rac{1cdot 3cdot 5(a/x)^7}{2cdot 4cdot 6cdot 7cdot 7} + cdotcdotcdot $
27.
$displaystyleintdisplaystyle rac{sec^{-1}(x/a)}{x^2}dx=left{ egin{arra... ...ystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight. $
28.
$displaystyleintcsc^{-1}displaystyle rac{x}{a}dx=left{ displaystyleeg... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight. $
29.
$displaystyleint xcsc^{-1}displaystyle rac{x}{a}dx=left{ egin{array}{l... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight. $

30.
$displaystyleint x^2csc^{-1}displaystyle rac{x}{a}dx=left{ egin{array}... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight. $
31.
$displaystyleintdisplaystyle rac{csc^{-1}(x/a)}{x}dx=-left(displaystyle ... ...{1cdot 3cdot 5(a/x)^7}{2cdot 4cdot 6cdot 7cdot 7}+cdotcdotcdot ight)$
32.
$displaystyleintdisplaystyle rac{csc^{-1}(x/a)}{x^2}dx=left{ egin{arra... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight. $
33.
$displaystyleint x^msin^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...e rac{1}{m+1}intdisplaystyle rac{x^{m+1}}{displaystyle sqrt{a^2-x^2}}dx$
34.
$displaystyleint x^mcos^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...e rac{1}{m+1}intdisplaystyle rac{x^{m+1}}{displaystyle sqrt{a^2-x^2}}dx$
35.
$displaystyleint x^m an^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...ac{x}{a}-displaystyle rac{a}{m+1}intdisplaystyle rac{x^{m+1}}{x^2+a^2}dx$
36.
$displaystyleint x^mcot^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...ac{x}{a}+displaystyle rac{a}{m+1}intdisplaystyle rac{x^{m+1}}{x^2+a^2}dx$
37.
$displaystyleint x^msec^{-1}displaystyle rac{x}{a}=left { egin{array}{... ...aystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight. $
38.
$displaystyleint x^mcsc^{-1}displaystyle rac{x}{a}dx=left{ egin{array}... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight. $


INTEGRALS CONTAINING eax

1.
$displaystyleint e^{ax} dx =displaystyle rac{e^{ax}}{a}$
2.
$displaystyleint xe^{ax}dx=displaystyle rac{e^{ax}}{a}left(x-displaystyle rac{1}{a} ight)$
3.
$displaystyleint x^2 e^{ax}dx=displaystyle rac{e^{ax}}{a}left(x^2-displaystyle rac{2x}{a}+displaystyle rac{2}{a^2} ight)$
4.
$egin{array}{lcl} displaystyleint x^n e^{ax}dx&=& displaystyle rac{x^n e^... ...2}}{a^2}-cdotcdotcdot displaystyle rac{(-1)^n n!}{a^n} ight) end{array}$
5.
$displaystyleintdisplaystyle rac{e^{ax}}{x}dx=ln x+displaystyle rac{ax}... ... rac{(ax)^2}{2cdot 2!}+displaystyle rac{(ax)^3}{3cdot 3!}+cdotcdotcdot$
6.
$displaystyleintdisplaystyle rac{e^{ax}}{x^n}dx=displaystyle rac{-e^{ax}... ...)x^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{e^{ax}}{x^{n-1}}dx$
7.
$displaystyleintdisplaystyle rac{dx}{p+qe^{ax}}=displaystyle rac{x}{p}-displaystyle rac{1}{ap}ln (p+qe^{ax})$
8.
$displaystyleintdisplaystyle rac{dx}{(p+qe^{ax})^2}=displaystyle rac{x}{... ...displaystyle rac{1}{ap(p+qe^{ax})}-displaystyle rac{1}{ap^2}ln(p+qe^{ax})$
9.
$displaystyleintdisplaystyle rac{dx}{pe^{ax}+qe^{-ax}}=left{ egin{array... ...style sqrt{-q/p}}{e^{ax}+displaystyle sqrt{-q/p}} ight) end{array} ight. $
10.
$displaystyleint e^{ax}sin bx dx=displaystyle rac{e^{ax}(asin bx -bcos bx)}{a^2+b^2}$
11.
$displaystyleint e^{ax}cos bx dx=e^{ax}displaystyle rac{(acos bx+bsin bx)}{a^2+b^2}$
12.
$displaystyleint xe^{ax}sin bx dx=displaystyle rac{xe^{ax}(asin bx -bcos... ...splaystyle rac{e^{ax}left{(a^2-b^2)sin bx-2abcos bx ight}}{(a^2+b^2)^2}$
13.
$displaystyleint xe^{ax}cos bx dx=displaystyle rac{xe^{ax}(acos bx +bsin... ...splaystyle rac{e^{ax}left{(a^2-b^2)cos bx+2absin bx ight}}{(a^2+b^2)^2}$
14.
$displaystyleint e^{ax}ln xdx=displaystyle rac{e^{ax}ln x}{a}-displaystyle rac{1}{a}intdisplaystyle rac{e^{ax}}{x}dx$
15.
$displaystyleint e^{ax}sin^n bxdx=displaystyle rac{e^{ax}sin^{n-1}bx}{a^2... ...cos bx) + displaystyle rac{n(n-1)b^2}{a^2+n^2b^2}int e^{ax}sin^{n-2}bx dx$
16.
$displaystyleint e^{ax}cos^n bxdx=displaystyle rac{e^{ax}cos^{n-1}bx}{a^2... ...sin bx) + displaystyle rac{n(n-1)b^2}{a^2+n^2b^2}int e^{ax}cos^{n-2}bx dx$


INTEGRALS CONTAINING ln(ax)

1.
$displaystyleint ln xdx=xln x-x$
2.
$displaystyleint xln x dx=displaystyle rac{x^2}{2}(ln x-displaystyle rac{1}{2})$
3.
$displaystyleint x^mln xdx=displaystyle rac{x^{m+1}}{m+1}left(ln x-displaystyle rac{1}{m+1} ight)$
4.
$displaystyleintdisplaystyle rac{ln x}{x}dx=displaystyle rac{1}{2}ln^2 x$
5.
$displaystyleintdisplaystyle rac{ln x}{x^2}dx=-displaystyle rac{ln x}{x}-displaystyle rac{1}{x}$
6.
$displaystyleintln^2 xdx=xln^2 x-2xln x+2x$
7.
$displaystyleintdisplaystyle rac{ln^n xdx}{x}=displaystyle rac{ln^{n+1}x}{n+1}$
8.