Mathematics

Prove $\underset{0}{\overset{\frac{\pi}{2}}{\int}} \log (\sin \, x) dx = - \dfrac{\pi}{2} \log \, 2$

SOLUTION
$I=\int_{0}^{\frac{\pi}{2}}\log (\sin x)dx$

$I=\int_{0}^{\frac{\pi}{2}}\log (\cos x)dx$

$2I=\int_{0}^{\frac{\pi}{2}}[\log (\sin x)+\log (\cos x)]dx$

$2I=\int_{0}^{\frac{\pi}{2}}\log (\sin x \cos x)dx$

$2I=\int_{0}^{\frac{\pi}{2}}\log \left ( \dfrac{\sin 2x}{2} \right )dx$

$2I=\int_{0}^{\frac{\pi}{2}} \log (\sin 2x)dx-\int_{0}^{\frac{\pi}{2}} \log 2 dx$

$2I=I_1-\log 2 [x]_0^{\frac{\pi}{2}}$

$2I=I_1-\dfrac{\pi}{2}\log 2$...(1)

$I_1=\int_{0}^{\frac{\pi}{2}} \log (\sin 2x)dx$

substitute $2x=t\rightarrow dt=2dx\Rightarrow x=0-\frac{\pi}{2}\rightarrow t=0-\pi$

$\therefore I_1=\dfrac{1}{2}\int_{0}^{\pi}\log (\sin t)dt$

$\Rightarrow I_1=I$...(2)

Placing (2)  in (1)

$\Rightarrow 2I=I-\dfrac{\pi}{2}\log 2$

$\therefore I=-\dfrac{\pi}{2}\log 2$

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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 Hard
Let $f$ be a positive function. Let
${ I }_{ 1 }=\int _{ 1-k }^{ k }{ xf\left\{ x(1-x) \right\} } dx$
${ I }_{ 2 }=\int _{ 1-k }^{ k }{ f\left\{ x(1-x) \right\} } dx$
where $2k-1>0$. Then $\cfrac { { I }_{ 1 } }{ { I }_{ 2 } }$
• A. $2$
• B. $k$
• C. $1$
• D. $\cfrac { 1 }{ 2 }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate : $\displaystyle \int \dfrac{x}{x^{4}+x^{2}+1}\ dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Let $\displaystyle \int e^{x}\left \{ f\left ( x \right )-f'\left ( x \right ) \right \}dx=\phi \left ( x \right ).$ Then $\displaystyle \int e^{x}f\left ( x \right )dx$ is
• A. $\displaystyle \phi \left ( x \right )+e^{x}f\left ( x \right )$
• B. $\displaystyle \phi \left ( x \right )-e^{x}f\left ( x \right )$
• C. $\displaystyle \frac{1}{2}\left \{ \phi \left ( x \right )+e^{x}f'\left ( x \right ) \right \}$
• D. $\displaystyle \frac{1}{2}\left \{ \phi \left ( x \right )+e^{x}f\left ( x \right ) \right \}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\int { \cfrac { \sin { x } }{ { \left( 1+\cos { x } \right) }^{ 2 } } } dx$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$