Mathematics

Evaluate : $$\displaystyle \int\frac{\cos x}{\sin(x-\dfrac{\pi}{6})\sin(x+\dfrac{\pi}{6})}dx$$


ANSWER

$$\log \left | \displaystyle \frac {2\sin x-1}{2\sin x+1} \right |+C$$


SOLUTION
$$\int\dfrac{\cos x}{\sin(x-\dfrac{\pi}{6})\sin(x+\dfrac{\pi}{6})}dx$$

$$\int\dfrac{\cos x}{\sin^2x-\sin^2\dfrac {\pi}{6}}dx$$

Let $$\sin x=t$$

or $$ dt= \cos x dx $$

$$\therefore  I=\int\dfrac {dt}{t^2-\dfrac {1}{4}}$$

$$=\dfrac {1}{2 \dfrac{1}{2}}\log \left | \dfrac {t-\dfrac{1}{2}}{t+\dfrac{1}{2}}\right |+C$$

$$=\log \left | \dfrac {2t-1}{2t+1} \right |+C$$

$$=\log \left | \dfrac {2\sin x-1}{2\sin x+1} \right |+C$$  where $$t=\sin x$$

Hence, option 'A' is correct.
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Find $$\int x^2 + a^2$$ and evaluate $$\int \dfrac{1}{3 + 2x + x^2}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$\displaystyle \int[\sin(\log x)+\cos(\log x)]dx=$$
  • A. $$\displaystyle e^{x} \sin (\log x)+c$$
  • B. $$\displaystyle e^{x} \cos (\log x)+c$$
  • C. $$\displaystyle x\cos (\log x)+c$$
  • D. $$\displaystyle x\sin (\log x)+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Hard
Evaluate:
$$\displaystyle \int { \cfrac { { x }^{ 2 } }{ { x }^{ 4 }-{ x }^{ 2 }-12 }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the integral:
$$\displaystyle\int \dfrac{-\sin x }{5+\cos x}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
$$\displaystyle \frac{3x-1}{(1-x+x^{2})(2+x)}=$$
  • A. $$\displaystyle \frac{x}{x^{2}-x+1}+\frac{1}{x+2}$$
  • B. $$\displaystyle \frac{x}{x^{2}+x+1}+\frac{2}{x+2}$$
  • C. $$\displaystyle \frac{x}{ -x+1}-\frac{2}{x+2}$$
  • D. $$\displaystyle \frac{x}{x^{2}-x+1}-\frac{1}{x+2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer