Integrate the following with respect to $x$: $\dfrac{x^{2}-3x+1}{\sqrt{1-x^{2}}}$

$-\dfrac{1}{2}x\sqrt{1-x^{2}}+3\sqrt{1-x^{2}}+\dfrac{3}{2}\sin^{-1}x+C$

CBSE Class 12 · 2015 · Indefinite IntegralsVery Short Answer

Integrate the following with respect to

x

\frac{x ^{2} - 3 x + 1}{1 - x ^{2}}

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

Write

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

, so

\frac{x ^{2} - 3 x + 1}{1 - x ^{2}} = - 1 - x^{2} - \frac{3 x}{1 - x ^{2}} + \frac{2}{1 - x ^{2}} .

Integrate each standard piece:

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

\int \frac{x d x}{1 - x ^{2}} = - 1 - x^{2}

, and

\int \frac{d x}{1 - x ^{2}} = sin^{- 1} x

. Combining,

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

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

CBSE Class 12 Board · 2015 · Subjective