English
CBSECommerce (English Medium) Class 12
Question Papers2489
Textbook Solutions20216
MCQ Online Mock Tests42
Important Solutions18873
Concept Notes & Videos242
Time Tables23
Syllabus

Evaluate the integral by using substitution. ∫0π2sinx1+cos2xdx - Mathematics

Advertisements
Advertisements

Question

Evaluate the integral by using substitution.

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

Sum

Solution

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

Substituting cos x = t,

⇒ - sin x dx = dt

And x = 0, t = 1, x `= pi/2,` t = 0

Hence, `I = - int_1^0 1/(1 + t^2)`  dt

`= - [tan^-1 t]_1^0`

`= - [tan^-1 0 - tan^-1 1]`

`= - [0 - pi/4]`

`= pi/4`

shaalaa.com
Evaluation of Definite Integrals by Substitution
  Is there an error in this question or solution?
Q 5
Chapter 7: Integrals - Exercise 7.10 [Page 340]

APPEARS IN

NCERT Mathematics [English] Class 12
Chapter 7 Integrals
Exercise 7.10 | Q 5 | Page 340

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

 


Evaluate: `int1/(xlogxlog(logx))dx`


Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`


Evaluate : `int1/(3+5cosx)dx`


 

Evaluate `∫_0^(3/2)|x cosπx|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 :

`int_e^(e^2) dx/(xlogx)`


Evaluate the integral by using substitution.

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


Evaluate the integral by using substitution.

`int_0^2 xsqrt(x+2)`  (Put x + 2 = `t^2`)


Evaluate the integral by using substitution.

`int_(-1)^1 dx/(x^2 + 2x  + 5)`


Evaluate of the following integral: 

\[\int x^\frac{5}{4} dx\]

Evaluate of the following integral: 

\[\int\frac{1}{x^5}dx\]

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 : 

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

Evaluate:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

 


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:

\[\int\limits_{- 5}^0 f\left( x \right) dx, where\ f\left( x \right) = \left| x \right| + \left| x + 2 \right| + \left| x + 5 \right|\]

 


Evaluate each of the following integral:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}dx\]

 


Evaluate each of the following integral:

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

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{x^{11} - 3 x^9 + 5 x^7 - x^5 + 1}{\cos^2 x}dx\]

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{\tan^7 x}{\tan^7 x + \cot^7 x}dx\]

Evaluate the following integral:

\[\int_0^\pi x\sin x \cos^2 xdx\]

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:

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

 


Find: `int_  (3"x"+ 5)sqrt(5 + 4"x"-2"x"^2)d"x"`.


`int_(pi/5)^((3pi)/10) [(tan x)/(tan x + cot x)]`dx = ?


If `I_n = int_0^(pi/4) tan^n theta  "d"theta " then " I_8 + I_6` equals ______.


`int_0^1 sin^-1 ((2x)/(1 + x^2))"d"x` = ______.


Evaluate the following:

`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`


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


Share
Notifications

Englishहिंदीमराठी
Login
Create free account


      Forgot password?
Course
Use app×