∫ 2 X ( 2 X + 1 ) 2 D X - Mathematics

Question

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

Sum

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?

Q 7 Q 6 Q 8

Chapter 19: Indefinite Integrals - Exercise 19.03 [Page 23]

APPEARS IN

RD Sharma Mathematics [English] Class 12

Chapter 19 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/(3+5cosx)dx`

Evaluate `int_(-1)^2|x^3-x|dx`

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

Evaluate: `intsinsqrtx/sqrtxdx`

Evaluate the integral by using substitution.

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

Evaluate the integral by using substitution.

`int_(-1)^1 dx/(x^2 + 2x + 5)`

`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 of the following integral:

\[\int 3^x dx\]

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 :

\[\int\frac{e^{6 \log_e x} - e^{5 \log_e x}}{e^{4 \log_e x} - e^{3 \log_e x}}dx\]

Evaluate:

\[\int\frac{1}{a^x b^x}dx\]

Evaluate:

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

Evaluate:

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