Evaluate $\int e^{x}\left(\frac{1-\sin x}{1-\cos x}\right)dx$

CBSE Class 12 · 2023 · Indefinite IntegralsLong Answer

Evaluate

\int e^{x} (\frac{1 - s i n x}{1 - c o s x}) d x

Answer: $- e^{x} cot \frac{x}{2} + C$

Use

1 - cos x = 2 sin^{2} \frac{x}{2}

and

sin x = 2 sin \frac{x}{2} cos \frac{x}{2}

\frac{1 - sin x}{1 - cos x} = \frac{1}{2} cosec^{2} \frac{x}{2} - cot \frac{x}{2} .

Take

f (x) = - cot \frac{x}{2}

; then

f^{'} (x) = \frac{1}{2} cosec^{2} \frac{x}{2}

, so the integrand is

e^{x} {f (x) + f^{'} (x)}

and

\int e^{x} {f + f^{'}} d x = e^{x} f (x) + C .

- e^{x} cot \frac{x}{2} + C

CBSE Class 12 Board · 2023 · Subjective