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

$$I=\dfrac{\pi }{2} \log 2$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\displaystyle \frac {\pi}{12}ln 2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Find $$\displaystyle  \int_{0}^{\pi/2} \sin x. \sin 2x\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer