IIT JEE , AIEEE Forum - Mathematics , Integral Calculus - can somebody pls post all the forms of indefinite integrals here?

iit_targetted

can somebody pls post all the forms of indefinite integrals here?

ramyani

This was given by NIVEDH
1.: $\displaystyle \int adx=ax$
2.: $\displaystyle \int af(x)dx=a \displaystyle \int f(x)dx$
3.: $\displaystyle \int \left( u \pm v \pm w \pm \cdots \right) dx = \displaystyle \int udx \pm \displaystyle \int vdx \pm \displaystyle \int wdx \pm \cdots$
4.: $\displaystyle \int udv = uv - \displaystyle \int vdu$
5.: $\displaystyle \int f(ax)dx = \displaystyle \frac{1}{a} \displaystyle \int f(u)du$
6.: $\displaystyle \int F\{f(x)\}dx = \displaystyle \int F(u) \displaystyle \frac{dx}{du}du = \displaystyle \int \displaystyle \frac{F(u)}{f'(x)}du$
7.: $\displaystyle \int u^{n}du = \displaystyle \frac{u^{n+1}}{n+1}, n \neq -1$
8.: $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{du}{u} & = & \ln u \mb... ...or} \ln (-u) \mbox{ if} u<0 \\ & = & \ln \left\vert u \right\vert \end{array}$
9.: $\displaystyle \int e^{u}du=e^{u}$
10.: $\displaystyle \int a^{u}du = \int e^{u \ln a}du = \displaystyle \frac{e^{u \ln a}}{\ln a} = \displaystyle \frac{a^{u}}{\ln a} , a >0, a \neq 1$
11.: $\displaystyle \int \sin u du = -\cos u$
12.: $\displaystyle \int \cos u du = \sin u$
13.: $\displaystyle \int \tan u du = \ln \sec u = -\ln \cos u$
14.: $\displaystyle \int \cot u du = \ln \sin u$
15.: $\displaystyle \int \sec u du = \ln (\sec u + \tan u) = \ln \tan \left( \displaystyle \frac{u}{2} + \displaystyle \frac{\pi}{4} \right)$
16.: $\displaystyle \int \csc u du = \ln (\csc u - \cot u) = \ln \tan \displaystyle \frac{u}{2}$

17.: $\displaystyle \int \sec ^{2} u du = \tan u$
18.: $\displaystyle \int \csc ^{2} u du = -\cot u$
19.: $\displaystyle \int \tan ^{2} u du = \tan u - u$
20.: $\displaystyle \int \cot ^{2} u du = -\cot u - u$
21.: $\displaystyle \int \sin ^{2} u du = \displaystyle \frac{u}{2} - \displaystyle \frac{\sin 2u}{4} = \displaystyle \frac{1}{2} (u-\sin u \cos u)$
22.: $\displaystyle \int \cos ^{2} u du = \displaystyle \frac{u}{2} + \displaystyle \frac{\sin 2u}{4} = \displaystyle \frac{1}{2} (u+\sin u \cos u)$
23.: $\displaystyle \int \sec u \tan u du = \sec u$
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.: $\displaystyle \int \tanh u du = \ln \cosh u$
28.: $\displaystyle \int \coth u du = \ln \sinh u$
29.: $\displaystyle \int$ sech $u du = \sin ^{-1}(\tanh u )$ or $2\tan ^{-1}e^{u}$
30.: $\displaystyle \int$ csch $u du = \ln \tanh \displaystyle \frac{u}{2}$ or $-\coth ^{-1}e^{u}$
31.: $\displaystyle \int$ sech $^{2} u du = \tanh u$
32.: $\displaystyle \int$ csch ² u du =-coth u
33.: $\displaystyle \int\tanh ^{2} u du = u - \tanh u$
34.: $\displaystyle \int$ coth ² u du = u -coth u
35.: $\displaystyle \int\sinh ^{2} u du = \displaystyle \frac{\sinh 2u}{4} - \displaystyle \frac{u}{2} = \displaystyle \frac{1}{2}(\sinh u \cosh u- u)$
36.: $\displaystyle \int\cosh ^{2} u du = \displaystyle \frac{\sinh 2u}{4} + \displaystyle \frac{u}{2} = \displaystyle \frac{1}{2}(\sinh u \cosh u+ u)$
37.: $\displaystyle \int$ sech $u \tanh u du = -$ sech u
38.: $\displaystyle \int$ csch ucoth u du = -csch u
39.: $\displaystyle \int\displaystyle \frac{du}{u^{2}+a^{2}} = \displaystyle \frac{1}{a} \tan^{-1} \displaystyle \frac{u}{a}$
40.: $\displaystyle \int\displaystyle \frac{du}{u^{2} - a^{2}}= \displaystyle \frac{1... ...n \left( \displaystyle \frac{u - a}{u+a} \right) = - \displaystyle \frac{1}{a}$ coth $^{-1} \displaystyle \frac{u}{a} , u^{2}>a^{2}$
41.: $\displaystyle \int\displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{... ...= \displaystyle \frac{1}{a} \tanh ^{-1} \displaystyle \frac{u}{a} , u^{2}<a^{2}$
42.: $\displaystyle \int\displaystyle \frac{du}{\sqrt{a^{2}-u^{2}}} = \sin ^{-1} \displaystyle \frac{u}{a}$
43.: $\displaystyle \int\displaystyle \frac{du}{\sqrt{u^{2}+a^{2}}} = \ln \left( u+ \displaystyle\sqrt{u^{2} + a^{2}} \right)$ or $\sinh ^{-1} \displaystyle \frac{u}{a}$
44.: $\displaystyle \int\displaystyle \frac{du}{\sqrt{u^{2}-a^{2}}} = \ln \left( u + \displaystyle\sqrt{u^{2} - a^{2}} \right)$
45.: $\displaystyle \int\displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}} = \displaystyle \frac{1}{a} \sec ^{-1} \left\vert \displaystyle \frac{u}{a} \right\vert$
46.: $\displaystyle \int\displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=-\displaystyle \frac{1}{a} \ln \left( \displaystyle \frac{a+\sqrt{u^{2}+a^{2}}}{u} \right)$
47.: $\displaystyle \int\displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=-\displaystyle \frac{1}{a} \ln \left( \displaystyle \frac{a+\sqrt{a^{2}-u^{2}}}{u} \right)$
48.: $\displaystyle \int f^{(n)}gdx = f^{(n-1)}g - f^{(n-2)}g' + f^{(n-3)} g'' - \cdots (-1)^{n} \displaystyle \int fg^{(n)}dx$

iit_targetted

thanx but i said i need different forms in indefinite integral.like the integration of ax^2+bx+c etc.there are about 43/45 different forms.pls help some one...

ramyani

How about some tricks ?

There is a general method to find integration of sin 9x / sin x , sin 11x/ sin x etc.

we know,

_{[ ]}^{[ ]} sinx dx / sin x = x + C

_{[ ]}^{[ ]} sin 3x dx / sin x = x + sin 2x + C

_{[ ]}^{[ ]} sin 5x dx / sin x = x + sin 2x + sin 4x / 4 + C

therefore,

_{[ ]}^{[ ]} sin 7x dx / sin x = x + sin 2x + sin 4x / 2 + sin 6x / 3 + C

in this method, we can find integration of sin 9x / sin x , sin 11x/ sin x etc.

( courtesy : Chakrapani Dutta, my friend )

ramyani

if u want I can provide proof.

silverdoe

These are good tricks. What's the logic?

vnkt.swaroop

vibhav1991

do refer rdsharma .
u shall get lot of them

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