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

$\log\!\left(\dfrac{x^{2}+1}{x^{2}+2}\right)+C$

CBSE Class 12 · 2022 · Indefinite IntegralsShort Answer

Find

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

Answer: $lo g (\frac{x ^{2} + 1}{x ^{2} + 2}) + C$

Put

t = x^{2}

, so

d t = 2 x d x

\int \frac{d t}{( t + 1 ) ( t + 2 )} .

Partial fractions:

\frac{1}{( t + 1 ) ( t + 2 )} = \frac{1}{t + 1} - \frac{1}{t + 2}

, so

lo g (t + 1) - lo g (t + 2) + C = lo g \frac{t + 1}{t + 2} + C .

Restoring

t = x^{2}

lo g (\frac{x ^{2} + 1}{x ^{2} + 2}) + C

CBSE Class 12 Board · 2022 · Subjective