Find:
      \int \frac{\cos x}{\left ( 1+\sin x \right )\left ( 2+\sin x \right )}dx

 

 

 

 
 
 
 
 

Answers (1)

I= \int \frac{\cos x}{\left ( 1+\sin x \right )\left ( 2+\sin x \right )}dx
Let \sin x= t
\cos x= \frac{dt}{dx}\Rightarrow \cos xdx= dt
I= \int \frac{dt}{\left ( 1+t \right )\left ( 2+t \right )}
Let \frac{1}{\left ( 1+t \right )\left ( 2+t \right )}= \frac{A}{1+t}+\frac{B}{2+t}= \frac{A\left ( 2+t \right )+B\left ( 1+t \right )}{\left ( 1+t \right )\left ( 2+t \right )}
                             1= \left ( 2+t \right )A+\left ( 1+t \right )B
t= -1\Rightarrow 1= A\left ( 1 \right )\Rightarrow A= 1
t= -2\Rightarrow 1= B\left ( -1 \right )\Rightarrow B=- 1
I= \int \frac{1}{\left ( 1+t \right )}-\frac{1}{\left ( 2+t \right )}dt
= \log \left ( 1+t \right )-\log \left ( 2+t \right )+C
I= \log\left | 1+\sin x \right |-\log \left | 2+\sin x \right |+C
I= \log\left | \frac{1+\sin x}{2+\sin x} \right |+C

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