DEFINITE INTEGRALS THAT CONTAIN TRIGONOMETRIC FUNCTIONS

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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.: $displaystyleint_{0}^{1}displaystyle rac{sin^{-1}x}{x}dx=displaystyle rac{pi}{2}ln2$

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DEFINITE INTEGRALS THAT CONTAIN TRIGONOMETRIC FUNCTIONS