March 3, 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 

Some Standard Elementary  Integrals

Integration Formulas

Important extension formulae of standard Integral forms

Integrals of some special functions

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

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

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

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

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

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

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

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

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

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

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

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

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

(xiv) \large \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

(xv) \large \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

(xvi) \large \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

Walli’s Formula of Integration

If m and n be integers, then

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

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

 

Class 12 integration formulas PDF

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

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

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