R.D. Sharma solutions for Mathematics Volume 1 and 2 [English] Class 12 chapter 19 - Definite Integrals [Latest edition]

\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]

Evaluate the following definite integrals:

\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]

Evaluate the following definite integral:

\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.

\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.

\[\int\limits_0^{( \pi )^{2/3}} \sqrt{x} \cos^2 x^{3/2} dx\]

\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

\[\int\limits_0^2 \left| x^2 - 3x + 2 \right| dx\]

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

\[\int_a^b \frac{x^\frac{1}{n}}{x^\frac{1}{n} + \left( a + b - x \right)^\frac{1}{n}}dx, n \in N, n \geq 2\]

If  \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that \[\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate the following integral:

Evaluate 

\[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\]

Evaluate the following integral:

Evaluate the following integral:

`int_0^(2a)f(x)dx`

Evaluate : 

If `f` is an integrable function such that f(2a − x) = f(x), then prove that

If f(2a − x) = −f(x), prove that

If f is an integrable function, show that

\[\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx\]

If f is an integrable function, show that

If f (x) is a continuous function defined on [0, 2a]. Then, prove that

If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that

If f(x) is a continuous function defined on [−a, a], then prove that 

Prove that:

Evaluate the following integrals as limit of sums:

Evaluate each of the following integral:

Evaluate each of the following  integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Evaluate each of the following integral:

Solve each of the following integral:

If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.

If \[\int\limits_0^a 3 x^2 dx = 8,\] write the value of a.

If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]

If \[\int_0^a \frac{1}{4 + x^2}dx = \frac{\pi}{8}\] , find the value of a.

Write the coefficient a, b, c of which the value of the integral

Evaluate : 

\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.  

If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:

\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\] equals

The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .

The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is 

The value of the integral \[\int\limits_0^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx\]  is 

\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\]  equals

\[\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx\] equals

\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\] equals

`int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"`

Given that \[\int\limits_0^\infty \frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)\left( x^2 + c^2 \right)} dx = \frac{\pi}{2\left( a + b \right)\left( b + c \right)\left( c + a \right)},\] the value of \[\int\limits_0^\infty \frac{dx}{\left( x^2 + 4 \right)\left( x^2 + 9 \right)},\]

The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]

The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is

If \[\int\limits_0^a \frac{1}{1 + 4 x^2} dx = \frac{\pi}{8},\] then a equals

If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals

The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is 

The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is

If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\]  then the value of I₁₀ + 90I₈ is

The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .

The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is

\[\int\limits_0^{2a} f\left( x \right) dx\]  is equal to

If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to

The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is

The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is 

The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is 

\[\int\limits_0^4 x\sqrt{4 - x} dx\]

\[\int\limits_1^2 x\sqrt{3x - 2} dx\]

\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]

\[\int\limits_0^1 \cos^{- 1} x dx\]

\[\int\limits_0^1 \tan^{- 1} x dx\]

\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]

\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]

\[\int\limits_0^{1/\sqrt{3}} \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]

\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]

\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]

\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx\]

\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]

\[\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx\]

\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]

\[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]

\[\int\limits_0^{\pi/4} \sin 2x \sin 3x dx\]

\[\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx\]

\[\int\limits_1^2 \frac{1}{x^2} e^{- 1/x} dx\]

\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]

\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx\]

\[\int\limits_0^{\pi/2} x^2 \cos 2x dx\]

\[\int\limits_0^1 \log\left( 1 + x \right) dx\]

Evaluate the following integrals :-

\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]

\[\int\limits_0^1 x \left( \tan^{- 1} x \right)^2 dx\]

\[\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx\]

\[\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx\]

\[\int\limits_0^{\pi/4} e^x \sin x dx\]

\[\int\limits_0^{\pi/4} \tan^4 x dx\]

\[\int\limits_0^1 \left| 2x - 1 \right| dx\]

\[\int\limits_1^3 \left| x^2 - 2x \right| dx\]

\[\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx\]

\[\int\limits_0^1 \left| \sin 2\pi x \right| dx\]

\[\int\limits_1^3 \left| x^2 - 4 \right| dx\]

\[\int\limits_{- \pi/2}^{\pi/2} \sin^9 x dx\]

\[\int\limits_{- 1/2}^{1/2} \cos x \log\left( \frac{1 + x}{1 - x} \right) dx\]

\[\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx\]

\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx\]

\[\int\limits_0^{2\pi} \cos^7 x dx\]

\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]

\[\int\limits_0^{\pi/2} \frac{1}{1 + \tan^3 x} dx\]

\[\int\limits_0^\pi \frac{x \sin x}{1 + \cos^2 x} dx\]

\[\int\limits_0^\pi x \sin x \cos^4 x dx\]

\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]

\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]

\[\int\limits_0^{15} \left[ x^2 \right] dx\]

\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]

\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]

\[\int\limits_0^{\pi/2} \frac{\cos^2 x}{\sin x + \cos x} dx\]

\[\int\limits_0^\pi \cos 2x \log \sin x dx\]

\[\int\limits_0^\pi \frac{x}{a^2 - \cos^2 x} dx, a > 1\]

\[\int\limits_0^\pi \frac{x \tan x}{\sec x + \tan x} dx\]

\[\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx\]

\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx\]

\[\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx\]

\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]

\[\int\limits_0^1 \cot^{- 1} \left( 1 - x + x^2 \right) dx\]

\[\int\limits_0^\pi \frac{dx}{6 - \cos x}dx\]

\[\int\limits_0^{\pi/2} \frac{1}{2 \cos x + 4 \sin x} dx\]

\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]

\[\int\limits_0^{\pi/2} \frac{dx}{4 \cos x + 2 \sin x}dx\]

\[\int\limits_0^4 x dx\]

\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]

\[\int\limits_1^4 \left( x^2 + x \right) dx\]

\[\int\limits_{- 1}^1 e^{2x} dx\]

\[\int\limits_2^3 e^{- x} dx\]

\[\int\limits_1^3 \left( 2 x^2 + 5x \right) dx\]

\[\int\limits_1^3 \left( x^2 + 3x \right) dx\]

\[\int\limits_0^2 \left( x^2 + 2 \right) dx\]

\[\int\limits_0^3 \left( x^2 + 1 \right) dx\]