Evaluate : $\int \frac{5x+3}{\sqrt{x^{2}+4x+10}}\,dx$

$5\sqrt{x^{2}+4x+10}-7\log\left|(x+2)+\sqrt{x^{2}+4x+10}\right|+C$

CBSE Class 12 · 2011 · Indefinite IntegralsLong Answer

Evaluate :

\int \frac{5 x + 3}{x ^{2} + 4 x + 10} d x

Answer: $5 x^{2} + 4 x + 10 - 7 lo g (x + 2) + x^{2} + 4 x + 10 + C$

Write

5 x + 3 = \frac{5}{2} (2 x + 4) - 7

, where

2 x + 4 = \frac{d}{d x} (x^{2} + 4 x + 10)

. Then

\int \frac{5 x + 3}{x ^{2} + 4 x + 10} d x = \frac{5}{2} \int \frac{( 2 x + 4 ) d x}{x ^{2} + 4 x + 10} - 7 \int \frac{d x}{x ^{2} + 4 x + 10} .

The first integral is

\frac{5}{2} \cdot 2 x^{2} + 4 x + 10 = 5 x^{2} + 4 x + 10

. For the second,

x^{2} + 4 x + 10 = (x + 2)^{2} + (6)^{2}

- 7 lo g (x + 2) + x^{2} + 4 x + 10 .

5 x^{2} + 4 x + 10 - 7 lo g (x + 2) + x^{2} + 4 x + 10 + C

CBSE Class 12 Board · 2011 · Subjective