Evaluate : $\int \frac{2x}{\left(x^{2}+1\right)\left(x^{2}+3\right)}\,dx$

$\dfrac{1}{2}\log\left|\dfrac{x^{2}+1}{x^{2}+3}\right|+C$

CBSE Class 12 · 2011 · Indefinite IntegralsLong Answer

Evaluate :

\int \frac{2 x}{( x ^{2} + 1 ) ( x ^{2} + 3 )} d x

Answer: $\frac{1}{2} lo g \frac{x ^{2} + 1}{x ^{2} + 3} + C$

Put

z = x^{2}

d z = 2 x d x

\int \frac{d z}{( z + 1 ) ( z + 3 )} .

Partial fractions:

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

, so

\frac{1}{2} lo g ∣ z + 1∣ - \frac{1}{2} lo g ∣ z + 3∣ + C = \frac{1}{2} lo g \frac{z + 1}{z + 3} + C .

Restoring

z = x^{2}

\frac{1}{2} lo g \frac{x ^{2} + 1}{x ^{2} + 3} + C

CBSE Class 12 Board · 2011 · Subjective