Find $\int \frac{x^{2}+x+1}{(x+2)\left(x^{2}+1\right)}\,dx$

$\dfrac{3}{5}\log|x+2|+\dfrac{1}{5}\log\left(x^{2}+1\right)+\dfrac{1}{5}\tan^{-1}x+C$

CBSE Class 12 · 2019 · Indefinite IntegralsLong Answer

Find

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

Answer: $\frac{3}{5} lo g ∣ x + 2∣ + \frac{1}{5} lo g (x^{2} + 1) + \frac{1}{5} tan^{- 1} x + C$

Resolve into partial fractions:

\frac{x ^{2} + x + 1}{( x + 2 ) ( x ^{2} + 1 )} = \frac{A}{x + 2} + \frac{B x + C}{x ^{2} + 1} .

Comparing coefficients gives

A = \frac{3}{5}, B = \frac{2}{5}, C = \frac{1}{5}

, so

\frac{3}{5} lo g ∣ x + 2∣ + \frac{1}{5} lo g (x^{2} + 1) + \frac{1}{5} tan^{- 1} x + C .

\frac{3}{5} lo g ∣ x + 2∣ + \frac{1}{5} lo g (x^{2} + 1) + \frac{1}{5} tan^{- 1} x + C

CBSE Class 12 Board · 2019 · Subjective