Mathematics

Find :- 
$$I = \displaystyle \int \dfrac{\sin \theta}{(4 \cos^2\theta)(2- \sin^2\theta)}d\theta $$


SOLUTION
$$\displaystyle I=\int \dfrac{\sin \theta d \theta }{(4\cos^{2} \theta )(2-\sin^{2} \theta )}$$
Let $$\cos  \theta =t$$
$$-\sin  \theta d \theta =dt$$
$$\sin  \theta .d \theta =-dt$$
$$=\displaystyle -\int \dfrac{dt}{4t^{2}(2-1+t^{2})}$$
$$=\displaystyle -\int \dfrac{dt}{4t^{2}(1+t^{2})}$$
$$=\displaystyle -\dfrac{1}{4}\int \dfrac{1+t^{2}-t^{2}}{t^{2}(1+t^{2})}$$
$$=\displaystyle -\dfrac{1}{4}\int \left[\frac{1+t^{2}}{t^{2}(1+t^{2})}-\frac{t^{2}}{t^{2}(1+t^{2})}\right]dt$$
$$=\displaystyle -\dfrac{1}{4}\int \left[\frac{1}{t^{2}}-\frac{1}{1+t^{2}}\right]dt$$
$$=-\dfrac{1}{4}\left[-\dfrac{1}{t}-\tan^{-1}(t)\right]+C$$
Put the value of $$t$$
$$=\displaystyle -\dfrac{1}{4}\left[\dfrac{-1}{\cos\theta }-\tan^{-1}(\cos\theta )\right]+C$$
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 Hard
Evaluate:
$$\int { \sqrt { { x }^{ 2 }-16 }  }  \ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
$$\displaystyle\int_{0}^{\pi/4}\dfrac{\tan^{3}x}{1+\cos 2x}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Hard
Evaluate the following integrals
$$\int { \cfrac { \cos { x } -\sin { x }  }{ \sqrt { 8-\sin { 2x }  }  }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard
Solve
$$ \int {\dfrac {x-2}{x ^2 - 4x + 3}} dx = $$
  • A. $$log \sqrt {x ^2 - 4x + 3 + c}$$
  • B. $$xlog (x - 3) - 2 log (x - 2) + c$$
  • C. $$log [(x - 3)(x - 1)]$$
  • D. $$None \ of \ these$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer