Formulas of Integration for Class 11 and 12 Mathematics

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Integration is the reverse process of differentiation.

By differentiation, we find the derivative of the given function, whereas by integration we find the function whose derivative is known. The function is called the integral of the given function.

The process of determining an integral of a function is called integration and the function to be integrated is called the integrand.

The letter x in dx denotes that the integration is to be performed with respect to the variable x.

Formula

(1) $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$ c is the integration constant

(2) $\int \frac{dx}{x}=log\left | x \right |+c$

(3) $\int e^{x}dx=e^{x}+c$ and $\int e^{mx}dx= \frac{e^{mx}}{m}+c$

(4) $\int a^{x}dx= \frac{a^{x}}{log_{e\, }a}+c$ and $\int a^{mx}dx= \frac{a^{mx}}{m\, log_{e\, }a}+c$

(5) $\int sin\, x\, dx=-cos\, x+c$ and $\int sin\, mx\, dx= -\, \frac{cos\, mx}{m}+c$

(6) $\int cos\, x\, dx= sin\, x+c$ and $\int cos\, mx\, dx=\frac{sin\, mx}{m}+c$

(7) $\int sec^{2\, }x\, dx=tan\, x+c$ and $\int sec^{2\, }mx\, dx=\frac{tan\, mx}{m}+c$

(8) $\int cosec^{2\, }x\, dx=-\, cot\, x+c$ and $\int cosec^{2\, }mx\, dx=-\, \frac{cot\, mx}{m}+c$

(9) $\int sec\, x\,\, tan\, x\,\, dx=sec\, x+c$ and $\int sec\, mx\, \, tan\, mx\, \, dx= \frac{sec\, mx}{m}+c$

(10) $\int cosec\, x\, \, cot\, x\, \, dx=-\, cosec\, x+c$ and $\int cosec\, mx\, \, cot\, mx\, \, dx=-\frac{cosec\, mx}{m}+c$

(11) $\int tan\, x\, \, dx=log\left | sec\, x \right |+c$ or $-\, log\left | cos\, x \right |+c$

(12) $\int cot\, x\, \, dx=log\left | sin\, x \right |+c$

(13) $\int sec\, x\, \, dx= log\left | sec\, x+tan\, x \right |+c$ or $log\left | tan\left ( \frac{\pi }{4}+\frac{x}{2} \right ) \right |+c$

(14) $\int cosec\, x\, \, dx=log\left | cosec\, x-cot\, x \right |+c$ or $log\left | tan\left ( \frac{x}{2} \right ) \right |+c$

(15) $\int \frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\, tan^{-1}\left ( \frac{x}{a} \right )+c$

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

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

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

(19) $\int \frac{dx}{\sqrt{x^{2}-a^{2}}}=log\left | x+\sqrt{x^{2}-a^{2}} \right |+c, \, \, \, \left | x \right |>a$

(20) $\int \frac{dx}{\sqrt{a^{2}-x^{2}}}=sin^{-1}\left ( \frac{x}{a} \right )+c$ , $\left | x \right |<a$

(21) $\int \frac{dx}{x\, \sqrt{x^{2}-a^{2}}}=\frac{1}{a}\,\, sec^{-1}\left ( \frac{x}{a} \right )+c$ , $x>a$

(22) $\int \sqrt{x^{2}+a^{2}}\,\, \, dx= \frac{x\, \sqrt{x^{2}+a^{2}}}{2}+\frac{a^{2}}{2}\:log\left | x+\sqrt{x^{2}+a^{2}} \right |+c$

(23) $\int \sqrt{x^{2}-a^{2}}\,\, \, dx= \frac{x\, \sqrt{x^{2}-a^{2}}}{2}-\frac{a^{2}}{2}\: log\left | x+\sqrt{x^{2}-a^{2}} \right |+c$

(24) $\int \sqrt{a^{2}-x^{2}}\, \, \, dx=\frac{x\, \sqrt{a^{2}-x^{2}}}{2}+\frac{a^{2}}{2}\, \, sin^{-1}\left ( \frac{x}{a} \right )+c$

INTEGRATION BY PARTS

$\int u\: v\: dx=u\: \int v\: dx-\int \left ( \frac{\mathrm{d} u}{\mathrm{d} x}\int v\: dx \right )\: dx+c$

In words: Integral of the product of two functions = first function × integral of second− integral of (derivative of first×integral of second).

Properties of definite Integrals:—-

(I) $\int_{a}^{b}f(x)\, dx=\int_{a}^{b}f(z)\, dz$

(II) $\int_{a}^{b}f(x)\, dx=-\, \int_{b}^{a}f(x)\, dx$

(III) $\int_{a}^{b}f(x)\, dx=\int_{a}^{c}f(x)\, dx+\int_{c}^{b}f(x)\, dx\: \mathrm{where}\: a<c<b$

(IV) $\int_{a}^{a}f(x)\, dx=0$

(V) $\int_{0}^{a}f(x)\, dx=\int_{0}^{a}f(a-x)\, dx$

(VI) (i) $\int_{0}^{2a}f(x)\, dx=\int_{0}^{a}f(x)\, dx+\int_{0}^{a}f(2a-x)\, dx$

(ii) $\int_{0}^{2a}f(x)\, dx=2\, \int_{0}^{a}f(x)\, dx,\, \, \mathrm{if}\: f(2a-x)=f(x)$

and

$\int_{0}^{2a}f(x)\, dx=0, \: \mathrm{if}\: f(2a-x)=-\, f(x)$

(VII) $\int_{-\, a}^{a}f(x)\, dx=2\, \int_{0}^{a}f(x)\, dx,\, \, \mathrm{if}\: f(x)\: \mathrm{is\, an\, even\, function\, of\,}\, x$

and

$\int_{-\, a}^{a}f(x)\, dx=0, \: \mathrm{if}\: f(x)\: \mathrm{is\, an\, odd\, function\, of\, }x$

(VIII) $\int_{a}^{b}f(x)\, dx=\int_{a}^{b}f(a+b-x)\, dx$

XI - XII M F Integration 1 XI - XII M F Integration 2 XI - XII M F Integration 3

ISC Class 12 Mathematics

Previous Years Board Questions	Solutions of Previous years Board Questions
Inverse Trigonometric Functions (2000 to 2020)	Inverse Trigonometric Functions (2000 to 2020)
Matrices (2000 to 2020)	Matrices (2000 to 2020)
Determinants (2000 to 2020)	Determinants (2000 to 2020)
Differentiation (2000 to 2020)	Differentiation (2000 to 2020)
Rolle’s Theorem and Lagrange’s Mean Value Theorem (1998 to 2020)	Rolle’s Theorem and Lagrange’s Mean Value Theorems (1998 to 2020)
Maxima and Minima (2000 to 2020)	Maxima and Minima (2000 to 2020)
Integration (1990 – 2011)	Integration (1990 – 2011)
Integration (2012 to 2020)	Integration (2012 to 2020)
Differential Equations (1991 to 2005)	Differential Equations (1991 to 2005)

Differential Equations (2006 to 2020)	Differential Equations (2006 to 2020)
Probability (Addition and Multiplication Theorem-Conditional Probability) (1993 to 2020)	Probability (Addition and Multiplication Theorem, Conditional Probability) (1993 to 2020)
Probability (Baye’s Theorem) (2000 to 2019)	Probability (Baye’s Theorem) (2000 to 2019)
Probability (Probability distribution and Binomial distribution) (1992 to 2020)	Probability (Probability distribution and Binomial distribution) (1992 to 2020)
Application of Calculus in Commerce and Economics (2005 to 2020)	Application of Calculus in Commerce and Economics (2005 to 2020)
Linear Programming (2008 to 2020)	Linear Programming (2008 to 2020)
Linear Regression (2005 to 2020)