Evaluate : $\int \frac{e^{x}}{\sqrt{5-4e^{x}-e^{2x}}}\,dx$

$\sin^{-1}\!\left(\dfrac{e^{x}+2}{3}\right)+C$

CBSE Class 12 · 2009 · Indefinite IntegralsLong Answer

Evaluate :

\int \frac{e ^{x}}{5 - 4 e ^{x} - e ^{2 x}} d x

Answer: $sin^{- 1} (\frac{e ^{x} + 2}{3}) + C$

Put

t = e^{x}

, so

d t = e^{x} d x

\int \frac{d t}{5 - 4 t - t ^{2}} .

Complete the square:

5 - 4 t - t^{2} = 9 - (t + 2)^{2} = 3^{2} - (t + 2)^{2}

, giving

\int \frac{d t}{3 ^{2} - ( t + 2 ) ^{2}} = sin^{- 1} (\frac{t + 2}{3}) + C .

Restoring

t = e^{x}

sin^{- 1} (\frac{e ^{x} + 2}{3}) + C

CBSE Class 12 Board · 2009 · Subjective