Mathematics

# Evaluate the following definite integral:$\displaystyle \int _{0}^{\pi /2} \cos^2 x\ dx$

##### SOLUTION

$I=\displaystyle \int _{0}^{\pi /2} \cos^2 x\ dx$

$=\displaystyle \int _{0}^{\pi /2} \dfrac {1+\cos 2x}{2} dx$  [$\because \cos^2 x=\dfrac {1+\cos 2x}{2}$]

$=\dfrac {1}{2} \left [x +\dfrac {\sin 2x}{2} \right]_0^{\pi /2}$

$=\dfrac {1}{2}\left [\left (\dfrac {\pi}{2}+\dfrac {\sin \pi}{2} \right) -\left (0+\dfrac {\sin 0}{2}\right) \right]$

$=\dfrac {\pi}{4}$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium

Evaluate the following definite integral:

$\displaystyle\int_{0}^{\pi/2}\dfrac{\sin^{3/2} x}{\sin^{3/2}x+\cos^{3/2} x}\ dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\int _{ -1 }^{ 3 }{ \left| { x }^{ 2 }-1 \right| dx }$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Solve:$\displaystyle \int \tan^6\theta d \theta$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int {\dfrac{{dx}}{{4{{\sin }^2}x + 4\sin x\cos x + 5{{\cos }^2}x}}}$ is equal to
• A. ${\tan ^{ - 1}}\left( {\tan x + \dfrac{1}{2}} \right) + c$
• B. $4{\tan ^{ - 1}}\left( {\tan x + \dfrac{1}{2}} \right) + c$
• C. none of these
• D. $\dfrac{1}{4}{\tan ^{ - 1}}\left( {\tan x + \dfrac{1}{2}} \right) + c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Subjective Medium
$\displaystyle\int\limits_{a}^{b}f(x)\ dx=b^3-a^3$, then find $f(x)$.

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020