Put \(1 + sinx = t\)
D.B.S:-
\(\cos x \,dx = dt\)
\(I = \int \frac {dt}{t(1 + t)}\)
\(= \int \frac {(1 + t )- (t)}{t(1 + t)} dt\)
\(= \int \frac 1t dt - \int \frac 1{1 + t}dt\)
\(= \ln(t) - \ln(1 + t) + C\)
\(=\ln|1 + \sin x|- \ln|1 + 1 + \sin x| + C\)
\(=\ln|1 + \sin x| -\ln|2 + \sin x|+ C\)
\(= \ln\left|\frac{1+\sin x}{2 + \sin x}\right| + C\)
