September 11, 2024

# [PDF] Integration Formulas for Class 12

Integration is the inverse process of differentiation. In differentiation, we are given a function and asked to determine its differential coefficient or derivative. In integration, we are given the derivative of a function and asked to find that function.

Here, fundamental integration formulas for various functions are mentioned. In addition to the fundamental integration formulas, this article also provides a classification of integral formulas.

Integration formulas for class 12

Contents

# Some Standard Elementary  Integrals

## Important extension formulae of standard Integral forms

### Integrals of some special functions

(i) $\large&space;\int&space;\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}log\left&space;|&space;\frac{x-a}{x+a}&space;\right&space;|+c$

(ii) $\large&space;\int&space;\frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}log\left&space;|&space;\frac{a+x}{a-x}&space;\right&space;|+c$

(iii) $\large&space;\int&space;\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}&space;tan^{-1}\frac{x}{a}+c$

(iv) $\large&space;\int&space;\frac{dx}{\sqrt{x^{2}-a^{2}}}&space;=log\left&space;|&space;x+&space;\sqrt{x^{2}-a^{2}}&space;\right&space;|+c$

(v) $\large&space;\int&space;\frac{dx}{\sqrt{a^{2}-x^{2}}}&space;=&space;sin^{-1}\frac{x}{a}+c$

(vi) $\large&space;\int&space;\frac{dx}{\sqrt{x^{2}+a^{2}}}=log\left&space;|&space;x+&space;\sqrt{x^{2}+a^{2}}\right&space;|+c$

(vii) $\large&space;\int&space;\frac{dx}{x\sqrt{x^{2}-a^{2}}}&space;=\frac{1}{a}sec^{-1}\frac{x}{a}+c$

(viii) $\large&space;\int&space;tanx&space;\,&space;dx&space;=&space;log&space;\left&space;|&space;secx&space;\right&space;|&space;+c$

(ix) $\large&space;\int&space;cot&space;x&space;\,&space;dx&space;=&space;log&space;\left&space;|&space;sinx&space;\right&space;|&space;+c$

(x) $\large&space;\int&space;secx&space;\,&space;dx&space;=&space;log&space;\left&space;|&space;(secx&space;+tanx)\right&space;|&space;+c&space;=log&space;\left&space;|&space;tan(\frac{\Pi&space;}{4}+\frac{x}{2})&space;\right&space;|+c$

(xi) $\large&space;\int&space;cosecx&space;\,&space;dx&space;=&space;log&space;\left&space;|&space;(cosecx&space;-&space;cotx)\right&space;|&space;+c&space;=log&space;\left&space;|&space;tan\frac{x}{2}&space;\right&space;|+c$

(xii) $\large&space;\int&space;e^{ax}sinbxdx=\frac{e^{ax}}{a^{2}+b^{2}}&space;\left&space;(&space;asinbx-bcosbx&space;\right&space;)+c$

(xiii)$\large&space;\int&space;e^{ax}cosbxdx=\frac{e^{ax}}{a^{2}+b^{2}}&space;\left&space;(&space;acosbx-bsinbx&space;\right&space;)+c$

(xiv) $\large&space;\int&space;\sqrt{a^{2}-x^{2}}\,&space;dx&space;=\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}sin^{-1}\frac{x}{a}+c$

(xv) $\large&space;\int&space;\sqrt{x^{2}+a^{2}}\,&space;dx&space;=\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}log\left&space;|&space;x+&space;\sqrt{x^{2}+a^{2}}&space;\right&space;|+c$

(xvi) $\large&space;\int&space;\sqrt{x^{2}-a^{2}}\,&space;dx&space;=\frac{x}{2}\sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2}log\left&space;|&space;x+&space;\sqrt{x^{2}-a^{2}}&space;\right&space;|+c$

#### Walli’s Formula of Integration

If m and n be integers, then

(i) $\large&space;\int_{0}^{\frac{\pi&space;}{2}}sin^{m}xcos^{n}xdx=\left\{\begin{matrix}&space;\frac{(m-1)(m-2)....(1\,&space;or\,&space;2).(n-1)(n-2).....(1\,&space;or\,&space;2)}{(m+n)(m+n-2).....(1\,&space;or\,&space;2)}\frac{\pi&space;}{2}\,&space;\,&space;where\,&space;m,&space;n&space;\,&space;are&space;\,&space;even\,&space;integers&space;\\&space;\frac{(m-1)(m-3)....(1\,&space;or\,&space;2).(n-1)(n-3).....(1\,&space;or\,&space;2)}{(m+n)(m+n-2).....(1\,&space;or\,&space;2)}\,&space;\,&space;where\,&space;m,&space;n&space;\,&space;are&space;\,&space;odd\,&space;integers&space;\end{matrix}\right.$

(ii) $\large&space;\int_{0}^{\frac{\pi&space;}{2}}sin^{n}xdx=\int_{0}^{\frac{\pi&space;}{2}}cos^{n}xdx=\left\{\begin{matrix}&space;\frac{(n-1)(n-3)(n-5)....&space;2}{(n)(n-2)(n-4)......,3}\,&space;\,&space;where&space;\,&space;\,&space;n&space;\,&space;is&space;\,&space;odd&space;\\&space;\frac{(n-1)(n-3)(n-5)....&space;3&space;.1}{(n)(n-2)(n-4).......,4&space;.2}\frac{\pi&space;}{2}\,&space;\,&space;where&space;\,&space;n&space;\,&space;is&space;\,&space;even&space;\end{matrix}\right.$

##### Class 12 integration formulas PDF

File Name: list-of-basic-integrals.pdf

File Name: extension-formulae-of-standard-integral-forms.pdf