Find $\int \sqrt{3-2x-x^{2}}\,dx$

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

CBSE Class 12 · 2019 · Indefinite IntegralsShort Answer

Find

\int 3 - 2 x - x^{2} d x

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

Complete the square:

3 - 2 x - x^{2} = 4 - (x + 1)^{2}

. Put

t = x + 1

d t = d x

\int 2^{2} - t^{2} d t = \frac{t}{2} 4 - t^{2} + \frac{4}{2} sin^{- 1} \frac{t}{2} + C .

Restoring

t = x + 1

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

CBSE Class 12 Board · 2019 · Subjective