Mathematics

Solve
$$\displaystyle \int \dfrac {1}{\sin x(3+2\cos x)}$$


SOLUTION

We have,

$$I=\int{\dfrac{1}{\sin x\left( 3+2\cos x \right)}dx}$$


On multiplying and divide by $$\sin x$$

$$ I=\int{\dfrac{\sin x}{{{\sin }^{2}}x\left( 3+2\cos x \right)}dx} $$

$$ I=\int{\dfrac{\sin x}{\left( 1-{{\cos }^{2}}x \right)\left( 3+2\cos x \right)}dx} $$

$$ I=\int{\dfrac{\sin x}{\left( 1-\cos x \right)\left( 1+\cos x \right)\left( 3+2\cos x \right)}dx} $$


Let $$ \cos x=t $$

$$ -\sin xdx=dt $$


So,

$$ I=-\int{\dfrac{1}{\left( 1-t \right)\left( t+1 \right)\left( 3+2t \right)}dt} $$

$$ I=\int{\dfrac{1}{\left( t-1 \right)\left( 1+t \right)\left( 3+2t \right)}}dt $$


Now, using partial dfraction and we get,

$$ \int{\dfrac{1}{\left( t-1 \right)\left( t+1 \right)\left( 2t+3 \right)}}dt=\int{\left[ \dfrac{1}{10\left( t-1 \right)}-\dfrac{1}{2\left( t+1 \right)}+\dfrac{4}{5\left( 2t+3 \right)} \right]dt} $$

$$ \Rightarrow \dfrac{1}{10}\int{\dfrac{1}{\left( t-1 \right)}dt}-\dfrac{1}{2}\int{\dfrac{1}{\left( t+1 \right)}dt}+\dfrac{4}{5}\int{\dfrac{1}{\left( 2t+3 \right)}}dt $$

$$ \Rightarrow \dfrac{1}{10}\ln \left| t-1 \right|-\dfrac{1}{2}\ln \left| t+1 \right|+\dfrac{4}{5}\dfrac{\ln \left| 2t+3 \right|}{2}+C $$

$$ \Rightarrow \dfrac{1}{10}\ln \left| \cos x-1 \right|-\dfrac{1}{2}\left| \cos x+1 \right|+\dfrac{2}{5}\ln \left| 2\cos x+3 \right|+C $$


Hence, this is the answer.

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 Single Correct Medium
Evaluate $$\displaystyle \int e^{\sin x}\sin 2xdx$$
  • A. $$2e^{\sin x}(\sin x+1)+c$$
  • B. $$e^{\sin x}(\sin x+2)+c$$
  • C. $$e^{\sin x}(3\sin x -2)+c$$
  • D. $$e^{\sin x}(2\sin x-2)+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Find $$\int a^{x} \cdot e^{x} dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Hard
Evaluate : $$\int \dfrac { e ^ { 2 x } - 1 } { e ^ { 2 x } + 1 } d x$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Multiple Correct Medium
If $$\int \frac{x\, cos\,  \alpha+1 }{(x^2+2x\, cos\,  \alpha+1)^{3/2}}$$ $$dx= \frac{x}{\sqrt{f(x) + g(x)cos\, \alpha }}+c$$ then (more than one option is correct)
  • A. g(2) = 2
  • B. f(1) = 2
  • C. f(2) = 5
  • D. g(1) = 2

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Solve : $$\displaystyle \int \dfrac{2x}{(x^2 + 1)(x^2 + 3)}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer