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

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

CBSE Class 12 · 2017 · Indefinite IntegralsVery Short Answer

Find

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

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

Put

y = x^{2}

, so

d y = 2 x d x

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

Partial fractions give

\frac{1}{( y + 1 ) ( y + 2 ) ^{2}} = \frac{1}{y + 1} - \frac{1}{y + 2} - \frac{1}{( y + 2 ) ^{2}}

, so

\int [\frac{1}{y + 1} - \frac{1}{y + 2} - \frac{1}{( y + 2 ) ^{2}}] d y = lo g (y + 1) - lo g (y + 2) + \frac{1}{y + 2} + C .

Restoring

y = x^{2}

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

CBSE Class 12 Board · 2017 · Subjective