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,

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

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

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

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

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

#### 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. $e^{\sin x}(2\sin x-2)+c$

1 Verified Answer | Published on 17th 09, 2020

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

1 Verified Answer | Published on 17th 09, 2020

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

1 Verified Answer | Published on 17th 09, 2020

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

Solve : $\displaystyle \int \dfrac{2x}{(x^2 + 1)(x^2 + 3)}$