Evaluate : $\int e^{x}\left(\frac{\sin 4x-4}{1-\cos 4x}\right)dx$

CBSE Class 12 · 2010 · Indefinite IntegralsLong Answer

Evaluate :

\int e^{x} (\frac{s i n 4 x - 4}{1 - c o s 4 x}) d x

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

Use

sin 4 x = 2 sin 2 x cos 2 x

and

1 - cos 4 x = 2 sin^{2} 2 x

\frac{sin 4 x - 4}{1 - cos 4 x} = \frac{2 sin 2 x cos 2 x - 4}{2 sin ^{2} 2 x} = cot 2 x - 2 cosec^{2} 2 x .

Take

f (x) = cot 2 x

; then

f^{'} (x) = - 2 cosec^{2} 2 x

, 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 2 x + C

CBSE Class 12 Board · 2010 · Subjective