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

 

 

 

 
 
 
 
 

Answers (1)

\int\frac{\cos x}{(1+ \sin x)(2 + \sin x)}dx

Let I =\int\frac{\cos x}{(1+ \sin x)(2 + \sin x)}dx

Put \sin x = t

Differentiating w.r.t x

\therefore \cos x dx = dt

\therefore I = \int \frac{dt}{(1+t)(2+ t)}

Let \frac{1}{(1+t)(2+ t)} = \frac{A}{1 + t} + \frac{B}{2 +t}

            1 = A(2 + t) + B(1+t)

Comparing coefficient of t

0 = A + B

A = - B\qquad -(i)

Comparing constants terms

\\1 = 2A + B\\ 1= -2B + B \\-1 = B

\therefore A = 1

\frac{1}{(1+t)(2+ t)} = \frac{1}{1 + t} - \frac{1}{2 +t}

Now, 

I=\int \frac{dt}{1 + t} - \int \frac{dt}{2 +t}

I = \log |1 +t| - \log |2 +t| + c

    = \log \left |\frac{1+t}{2+ t}\right | + c \qquad \left[\because \log m - \log n = \log \left(\frac{m}{n} \right )\right]

I\Rightarrow \log \left |\frac{1+\sin x}{2+ \sin x}\right | + c

 

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