Mathematics

# $\displaystyle\int\limits_0^{\frac{\pi }{2}} {\left( {2\log \sin x - \log \sin 2x} \right)dx}$

##### SOLUTION
$I=\displaystyle\int_{0}^{\dfrac{\pi }{2}}(2 \log \sin x- \log \sin 2x)dx$

we have $f(x)=2 \log \sin x- \log \sin 2x$

$f(x)=\log \dfrac{\sin^{2}x}{\sin 2x}$

$=\log \tan x-\log 2$

$I_{1}=\displaystyle\int_{0}^{\dfrac{\pi }{2}}\log \tan x\ dx$

Let $y=\dfrac{\pi }{2}-x, dy=-dx, \tan x= \cot y$

$I_{1}= -\displaystyle\int_{0}^{\dfrac{\pi }{2}}\log \cot y \ dy=-\int_{0}^{\dfrac{\pi }{2}}\log \tan x\ dx=-I$

$\Rightarrow 2I_{1}=0$

$\Rightarrow I_{1}=0$

Now $I= \displaystyle\int_{0}^{\dfrac{\pi }{2}}\log \tan x dx-\int_{0}^{\dfrac{\pi }{2}}\log 2 \ dx$

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

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

#### Realted Questions

Q1 Subjective Medium
Solve:
$\displaystyle \int x^{2} \sin^{2}x dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Medium
##### ASSERTION

$\displaystyle \int \frac{10x^{9}+10^{x}\log_{e}10}{10^{x}+x^{10}}dx=\log \left | 10^{x}+x^{10} \right |+C$

##### REASON

$\displaystyle \int \frac{{f}'\left ( x \right )}{f\left ( x \right )}dx=\log \left | f\left ( x \right ) \right |+C$

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The integral $\displaystyle \int_{0}^{1/2}\frac{\ln({1+2x)}}{1+4x^2}\mathrm{d}x$ equals :
• A. $\displaystyle \frac {\pi}{4}ln 2$
• B. $\displaystyle \frac {\pi}{8}ln 2$
• C. $\displaystyle \frac {\pi}{32}ln 2$
• D. $\displaystyle \frac {\pi}{12}ln 2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Find the integer closest to $\displaystyle \int_{0}^{2\pi}\dfrac{\pi dx}{(1+2^{sin\,x})(1+2^{cos\,x})}$.

Find $\displaystyle \int_{0}^{\pi/2} \sin x. \sin 2x\ dx$