Find $\int (x+3)\sqrt{3-4x-x^{2}}\,dx$

$-\dfrac{1}{3}\left(3-4x-x^{2}\right)^{3/2}+\dfrac{x+2}{2}\sqrt{3-4x-x^{2}}+\dfrac{7}{2}\sin^{-1}\!\left(\dfrac{x+2}{\sqrt7}\right)+C$

CBSE Class 12 · 2015 · Indefinite IntegralsLong Answer

Find

\int (x + 3) 3 - 4 x - x^{2} d x

Write

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

, where

- 2 x - 4 = \frac{d}{d x} (3 - 4 x - x^{2})

. Then

\int (x + 3) 3 - 4 x - x^{2} d x = - \frac{1}{2} \int (- 2 x - 4) 3 - 4 x - x^{2} d x + \int 3 - 4 x - x^{2} d x .

The first integral is

- \frac{1}{2} \cdot \frac{2}{3} (3 - 4 x - x^{2})^{3/2} = - \frac{1}{3} (3 - 4 x - x^{2})^{3/2}

. For the second,

3 - 4 x - x^{2} = 7 - (x + 2)^{2}

\int 7 - (x + 2)^{2} d x = \frac{x + 2}{2} 3 - 4 x - x^{2} + \frac{7}{2} sin^{- 1} (\frac{x + 2}{7}) .

- \frac{1}{3} (3 - 4 x - x^{2})^{3/2} + \frac{x + 2}{2} 3 - 4 x - x^{2} + \frac{7}{2} sin^{- 1} (\frac{x + 2}{7}) + C

CBSE Class 12 Board · 2015 · Subjective