Mathematics

# $\int { \sin ^{ 2 }{ \cfrac { x }{ 2 } } } dx\quad$

##### SOLUTION
$\displaystyle\int{{\sin}^{2}{\dfrac{x}{2}}dx}$

$=\dfrac{1}{2}\displaystyle\int{2{\sin}^{2}{\dfrac{x}{2}}dx}$

We know that $\cos{x}=1-2{\sin}^{2}{\dfrac{x}{2}}$

$=\dfrac{1}{2}\displaystyle\int{\left(1-\cos{x}\right)dx}$

$=\dfrac{1}{2}\left[x-\sin{x}\right]+c$

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

#### Realted Questions

Q1 Single Correct Medium
$\int 3\sqrt{x}7\sqrt{1+3\sqrt{x^{4}}}dx$ is equal to :
• A. $\dfrac{32}{21}\left(1+3\sqrt{x^{4}}\right)^{8/7}+C$
• B. $\dfrac{7}{32}\left(1+3\sqrt{x^{4}}\right)^{8/7}+C$
• C. $none\ of\ these$
• D. $\dfrac{21}{32}\left(1+3\sqrt{x^{4}}\right)^{8/7}+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int { \frac { \sin ^{ 3 }{ xdx } }{ \left( 1+\cos ^{ 2 }{ x } \right) \sqrt { 1+\cos ^{ 2 }{ x } +\cos ^{ 4 }{ x } } } dx }$ is equal to
• A. $\sec ^{ -1 }{ \left( \sec { x } -\cos { x } \right) } +c$
• B. $\sec ^{ -1 }{ \left( \cos { x } -\tan { x } \right) } +c$
• C. $\sec ^{ -1 }{ \left( \cos { x } +\tan { x } \right) } +c$
• D. $\sec ^{ -1 }{ \left( \sec { x } +\cos { x } \right) } +c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int {\dfrac{{{{\left( {x + 2} \right)}^2}}}{{\sqrt x }}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int_{\sin \theta }^{\cos \theta }f(x \tan \theta )dx\left ( where \theta \neq \frac{n\pi}{2} ,n\epsilon I\right )$ is equal to
• A. $\displaystyle -\tan \theta \int_{\sin \theta }^{\cos \theta }f(x)dx$
• B. $\displaystyle \sin \theta \int_{1}^{\tan }f(x\cos \theta )dx$
• C. $\displaystyle \frac{1}{\tan \theta }\int_{\sin \theta }^{\sin \theta \tan \theta}f(x)dx$
• D. $\displaystyle -\int_{1}^{\tan }f(x\sin \theta )dx$

$\int { \cfrac { 5x+8 }{ { x }^{ 2 }(3x+8) } } dx$