Question

Evaluate:

`∫(sin x  dx)/(cos x (1 − sin x))`

Answer 1

I = `∫ (sin x cos x)/((1 − sin^2 x)(1 − sin x)) dx`

I = `∫ (t)/((1 + t)(1 − t)^2) dt`   ...[Let sin x = t, then cos x dx = dt]

= `t/((1 + t)(1 − t)^2)`

= `(−1/4)/(1 + t) + (1/4)/(1 − t) + (1/2)/(1 − t)^2`   ...[Breaking the integrand into partial fractions]

I = `−1/4 1n|1 + t| + 1/4 1n |1 − t| + 1/(2(1 − t)) + C`   ...[Integrating each term separately]

I = `1/4 1n |(1 − t)/(1 + t)| + 1/(2(1 − t)) + C`   ...[substituting t = sin x back]

I `1/4 1n |(1 − sin x)/(1 + sin x)| + 1/(2(1 − sin x)) + C`