Integration class 12 revision topic

INDEFINITE INTEGRALS
IMPORTANT FORMULAS USED IN INTEGRALS

∫xndx=xn+1n+1+C$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$$

∫1xdx=logx+C$$\int \frac{1}{x}dx=logx+C$$

∫exdx=ex+C$$\int e^{x}dx=e^x+C$$

∫axdx=axloga+C$$\int a^{x}dx=\frac{a^x}{loga}+C$$

∫1dx=x+C$$\int 1dx=x+C$$

∫kdx=kx+C$$\int kdx=kx+C$$

∫0dx=C$$\int 0dx=C$$

∫[f(x)]n.f′(x)dx=[f(x)]n+1n+1+C$$\int [f(x){]}^n.f^{{}^{\prime }}(x)dx=\frac{[f(x){]}^{n+1}}{n+1}+C$$

∫f′(x)f(x)dx=log|f(x)|+C$$\int \frac{f^{{}^{\prime }}(x)}{f(x)}dx=log\left|f(x)\right|+C$$

INTEGRAL WITH TRIGONOMETRY

∫sinxdx=−cosx+C$$\int sinxdx=-cosx+C$$

∫cosxdx=sinx+C$$\int cosxdx=sinx+C$$

∫tanxdx=log|secx|+C,or−log|cosx|+C$$\int tanxdx=log\left|secx\right|+C,or-log\left|cosx\right|+C$$

∫cotxdx=log|sinx|+C,(or)−log|cosecx|+C$$\int cotxdx=log\left|sinx\right|+C,(or)-log\left|cosecx\right|+C$$

∫secxdx=log|secx+tanx|+C=log∣∣tan(π4+x2)∣∣+C$$\int secxdx=log\left|secx+tanx\right|+C=log\left|tan\left(\frac{\pi }{4}+\frac{x}{2}\right)\right|+C$$

∫cosecxdx=log|cosecx−cotx|+C=−log|cosecx+cotx|+C=log∣∣tanx2∣∣+C$$\int cosecxdx=log\left|cosecx-cotx\right|+C=-log\left|cosecx+cotx\right|+C=log\left|tan\frac{x}{2}\right|+C$$

∫sec2xdx=tanx+C$$\int se{c}^{2}xdx=tanx+C$$

∫cosec2xdx=−cotx+C$$\int cose{c}^{2}xdx=-cotx+C$$

∫secxtanxdx=secx+C$$\int secxtanxdx=secx+C$$

∫cosecxcotxdx=−cosecx+C$$\int cosecxcotxdx=-cosecx+C$$

∫1a2−x2−−−−−−√dx=sin−1(xa)+C=−cos−1(xa)+C$$\int \frac{1}{\sqrt{a^2-x^2}}dx=si{n}^{-1}\left(\frac{x}{a}\right)+C=-co{s}^{-1}\left(\frac{x}{a}\right)+C$$

∫1a2+x2dx=1atan−1(xa)+C=−1acot−1(xa)+C$$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C=-\frac{1}{a}{\cot }^{-1}\left(\frac{x}{a}\right)+C$$

∫1xx2−a2−−−−−−√dx=1asec−1(xa)+C=−1acosec−1(xa)+C$$\int \frac{1}{x\sqrt{x^2-a^2}}dx=\frac{1}{a}se{c}^{-1}\left(\frac{x}{a}\right)+C=-\frac{1}{a}cose{c}^{-1}\left(\frac{x}{a}\right)+C$$

Integration formulas Part 1 class 12-CBSE

INTEGRAL OF SOME SPECIAL CASES

∫dxa2−x2=12alog∣∣∣x+ax−a∣∣∣+C$$\int \frac{dx}{a^2-x^2}=\frac{1}{2a}log\left|\frac{x+a}{x-a}\right|+C$$

∫dxx2−a2=12alog∣∣∣x−ax+a∣∣∣+C$$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}log\left|\frac{x-a}{x+a}\right|+C$$

∫dxx2+a2=1atan−1ab+C$$\int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\frac{a}{b}+C$$

∫dxx2−a2−−−−−−√=log∣∣x+x2−a2−−−−−−√∣∣+C$$\int \frac{dx}{\sqrt{x^2-a^2}}=log\left|x+\sqrt{x^2-a^2}\right|+C$$

∫dxx2+a2−−−−−−√=log∣∣x+x2+a2−−−−−−√∣∣+C$$\int \frac{dx}{\sqrt{x^2+a^2}}=log\left|x+\sqrt{x^2+a^2}\right|+C$$

∫dxa2−x2−−−−−−√=sin−1xa+C$$\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\frac{x}{a}+C$$

∫x2−a2−−−−−−√dx=x2x2−a2−−−−−−√−a22log∣∣x+x2−a2−−−−−−√∣∣+C$$\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}log\left|x+\sqrt{x^2-a^2}\right|+C$$

∫x2+a2−−−−−−√dx=x2x2+a2−−−−−−√+a22log∣∣x+x2+a2−−−−−−√∣∣+C$$\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}log\left|x+\sqrt{x^2+a^2}\right|+C$$

∫a2−x2−−−−−−√dx=x2a2−x2−−−−−−√+a22sin−1xa+C$$\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}+C$$

Integration formulas Part 1 class 12-CBSE Mathematics

INTEGRATION BY PARTIAL FRACTION

px+q(x−a)(x−b)=Ax−a+Bx−b$$\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}$$

px+q(x−a)2=Ax−a+B(x−a)2$$\frac{px+q}{(x-a{)}^2}=\frac{A}{x-a}+\frac{B}{{\left(x-a\right)}^2}$$

px2+qx+r(x−a)2(x−b)=Ax−a+B(x−a)2+Cx−b$$\frac{p{x}^2+qx+r}{(x-a{)}^2\left(x-b\right)}=\frac{A}{x-a}+\frac{B}{{\left(x-a\right)}^2}+\frac{C}{x-b}$$

px2+qx+r(x−a)(x−b)(x−c)=Ax−a+B(x−b)+Cx−c$$\frac{p{x}^2+qx+r}{(x-a)\left(x-b\right)\left(x-c\right)}=\frac{A}{x-a}+\frac{B}{\left(x-b\right)}+\frac{C}{x-c}$$

px2+qx+r(x−a)(x2+bx+c)=Ax−a+Bx+Cx2+bx+c$$\frac{p{x}^2+qx+r}{\left(x-a\right)\left(x^2+bx+c\right)}=\frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}$$

INTEGRATION BY PARTS

∫uvdx=u∫vdx−∫(ddxu∫vdx)dx$$\int uvdx=u\int vdx-\int \left(\frac{d}{dx}u\int vdx\right)dx$$

Integration formulas Part 1 class 12-CBSE Mathematics

While using the Integration by parts the priority of taking 1st function is as follows

    I......Inverse Trigonometric Functions

    L.....Logarithmic Function

    A.....Algebraic Function(Polynomial Function

    T......Trigonometric Function

    E...... Exponential Function

Integration With Exponential Function

∫ex[f(x)+f′(x)]dx=exf(x)+C$$\int e^x\left[f(x)+f^{{}^{\prime }}(x)\right]dx=e^{x}f(x)+C$$$$$$
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$$By\; Shikha\; kaushal$$

∫f′(x)[f(x)]ndx=[f(x)]−n+1−n+1