Evaluate the integral by using substitution. ∫02dxx+4-x2

Question

Evaluate the integral by using substitution.

`int_0^2 dx/(x + 4 - x^2)`

Sum

Solution

Let `I = int_0^2 dx/(x + 4 - x^2)`

`= int_0^2 dx/(4 - (x^2 - x))`

`= int_0^2 dx/(4 + 1/4 - (x - 1/2)^2)`

`= int_0^2 dx/((sqrt17/2)^2 - (x - 1/2)^2)`

`= 1/(2 xx sqrt17/2) [log (sqrt17/2 + (x - 1/2))/(sqrt17/2 - (x - 1/2)}]_0^2`

`= 1/sqrt17 [log (sqrt17 + 2x - 1)/(sqrt17 - 2x + 1)]_0^2`

`= 1/sqrt17 [log (sqrt17 + 3)/(sqrt17 - 3) - log (sqrt17 - 1)/(sqrt17 + 1)]`

`= 1/sqrt17 log [(sqrt17 + 3)/(sqrt17 - 3) xx (sqrt17 + 1)/(sqrt17 - 1)]`

`= 1/sqrt17 log [(17 +3 + 3sqrt17 + sqrt17)/(17 + 3 - 3sqrt17 - sqrt17)]`

`= 1/sqrt17 log ((20 + 4sqrt17)/(20 - 4sqrt17))`

`= 1/sqrt17 log ((5 + sqrt17)/(5 - sqrt17))`

`= 1/sqrt17 log ((5 + sqrt17)/(5 - sqrt17) xx (5 + sqrt17)/(5 + sqrt17))`

`= 1/sqrt17 log [(25 + 17 + 10sqrt17)/(25 - 17)]`

`= 1/sqrt17 log [(41 + 10 sqrt17)/8]`

`= 1/sqrt17 log [(21 + 5 sqrt17)/4]`

shaalaa.com

Evaluation of Definite Integrals by Substitution

Is there an error in this question or solution?

Q 6 Q 5 Q 7

Chapter 7: Integrals - Exercise 7.10 [Page 340]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12

Chapter 7 Integrals
Exercise 7.10 | Q 6 | Page 340

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: `intsinsqrtx/sqrtxdx`

Evaluate the integral by using substitution.

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

Evaluate the integral by using substitution.

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

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 of the following integral:

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

Evaluate of the following integral:

\[\int \log_x \text{x dx}\]

Evaluate:

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

Evaluate:

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

Evaluate:

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

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

\[\int\limits_2^8 \left| x - 5 \right| dx\]

Evaluate the following integral:

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