Evaluate : $\int \frac{x+2}{\sqrt{x^{2}+5x+6}}\,dx$

$\sqrt{x^{2}+5x+6}-\dfrac{1}{2}\log\left|\left(x+\tfrac{5}{2}\right)+\sqrt{x^{2}+5x+6}\right|+C$

CBSE Class 12 · 2014 · Indefinite IntegralsLong Answer

Evaluate :

\int \frac{x + 2}{x ^{2} + 5 x + 6} d x

Answer: $x^{2} + 5 x + 6 - \frac{1}{2} lo g (x + \frac{5}{2}) + x^{2} + 5 x + 6 + C$

Write

x + 2 = \frac{1}{2} (2 x + 5) - \frac{1}{2}

, where

2 x + 5 = \frac{d}{d x} (x^{2} + 5 x + 6)

. Then

\int \frac{x + 2}{x ^{2} + 5 x + 6} d x = \frac{1}{2} \int \frac{( 2 x + 5 ) d x}{x ^{2} + 5 x + 6} - \frac{1}{2} \int \frac{d x}{x ^{2} + 5 x + 6} .

The first integral is

x^{2} + 5 x + 6

. For the second,

x^{2} + 5 x + 6 = (x + \frac{5}{2})^{2} - (\frac{1}{2})^{2}

- \frac{1}{2} lo g (x + \frac{5}{2}) + x^{2} + 5 x + 6 .

x^{2} + 5 x + 6 - \frac{1}{2} lo g (x + \frac{5}{2}) + x^{2} + 5 x + 6 + C

CBSE Class 12 Board · 2014 · Subjective