English

CBSECommerce (English Medium) Class 12

Question Papers

Question Papers2965

Textbook Solutions20276

MCQ Online Mock Tests42

Important Solutions23871

Concept Notes & Videos603

∫ 2 X ( 2 X + 1 ) 2 D X

Advertisements

Advertisements

Question

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

Sum

Advertisements

Solution

\[\int\frac{2x}{\left( 2x + 1 \right)^2}dx\]
\[ = \int\left( \frac{2x + 1 - 1}{\left( 2x + 1 \right)^2} \right)dx\]
\[ = \int\left[ \frac{2x + 1}{\left( 2x + 1 \right)^2} - \frac{1}{\left( 2x + 1 \right)^2} \right]dx\]
\[ = \int\frac{dx}{2x + 1} - \int \left( 2x + 1 \right)^{- 2} dx\]
\[ = \frac{\log\left( 2x + 1 \right)}{2} - \left[ \frac{\left( 2x + 1 \right)^{- 2 + 1}}{2\left( - 2 + 1 \right)} \right] + C\]
\[ = \frac{\log \left( 2x + 1 \right)}{2} + \frac{\left( 2x + 1 \right)^{- 1}}{2} + C\]
\[ = \frac{\log \left( 2x + 1 \right)}{2} + \frac{1}{2\left( 2x + 1 \right)} + C\]

shaalaa.com

Evaluation of Definite Integrals by Substitution

Is there an error in this question or solution?

Chapter 18: Indefinite Integrals - Exercise 19.03 [Page 23]

APPEARS IN

R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12

Chapter 18 Indefinite Integrals
Exercise 19.03 | Q 7 | Page 23

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Evaluation of Definite Integrals by Substitution

video tutorial00:18:12

RELATED QUESTIONS

Evaluate: `int1/(xlogxlog(logx))dx`

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

find `∫_2^4 x/(x^2 + 1)dx`

If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.

Evaluate the integral by using substitution.

`int_0^1 x/(x^2 +1)`dx

Evaluate the integral by using substitution.

`int_0^(pi/2) sqrt(sin phi) cos^5 phidphi`

The value of the integral `int_(1/3)^4 ((x- x^3)^(1/3))/x^4` dx is ______.

`int 1/(1 + cos x)` dx = _____

A) `tan(x/2) + c`

B) `2 tan (x/2) + c`

C) -`cot (x/2) + c`

D) -2 `cot (x/2)` + c

Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`

Evaluate of the following integral:

(i) \[\int x^4 dx\]

Evaluate of the following integral:

\[\int\frac{1}{\sqrt[3]{x^2}}dx\]

Evaluate of the following integral:

\[\int 3^{2 \log_3} {}^x dx\]

Evaluate:

\[\int\sqrt{\frac{1 - \cos 2x}{2}}dx\]

Evaluate:

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

Evaluate:

\[\int\frac{e\log \sqrt{x}}{x}dx\]

Evaluate the following definite integral:

\[\int_0^1 \frac{1}{\sqrt{\left( x - 1 \right)\left( 2 - x \right)}}dx\]

Evaluate the following integral:

\[\int\limits_{- 4}^4 \left| x + 2 \right| dx\]

Evaluate the following integral:

\[\int\limits_{- 2}^2 \left| x + 1 \right| dx\]

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

\[\int\limits_0^4 \left| x - 1 \right| dx\]

Evaluate the following integral:

\[\int\limits_1^4 \left\{ \left| x - 1 \right| + \left| x - 2 \right| + \left| x - 4 \right| \right\} dx\]

Evaluate the following integral:

\[\int\limits_0^4 \left( \left| x \right| + \left| x - 2 \right| + \left| x - 4 \right| \right) dx\]

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{\tan^2 x}{1 + e^x}dx\]

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{\cos^2 x}{1 + e^x}dx\]

Evaluate the following integral:

\[\int_{- \pi}^\pi \frac{2x\left( 1 + \sin x \right)}{1 + \cos^2 x}dx\]

Evaluate the following integral:

\[\int_{- 2}^2 \frac{3 x^3 + 2\left| x \right| + 1}{x^2 + \left| x \right| + 1}dx\]

Evaluate the following integral:

\[\int_0^{2\pi} \sin^{100} x \cos^{101} xdx\]

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{a\sin x + b\sin x}{\sin x + \cos x}dx\]

Find : \[\int e^{2x} \sin \left( 3x + 1 \right) dx\] .

Find : \[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\] .

Evaluate: `int_-1^2 (|"x"|)/"x"d"x"`.

`int_0^3 1/sqrt(3x - x^2)"d"x` = ______.

The value of `int_0^1 (x^4(1 - x)^4)/(1 + x^2) dx` is

Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.

Evaluate:

`int (1 + cosx)/(sin^2x)dx`

Notifications