Evaluate : $\int \frac{6x+7}{\sqrt{(x-5)(x-4)}}\,dx$

$6\sqrt{x^{2}-9x+20}+34\log\left|\left(x-\tfrac{9}{2}\right)+\sqrt{x^{2}-9x+20}\right|+C$

CBSE Class 12 · 2011 · Indefinite IntegralsLong Answer

Evaluate :

\int \frac{6 x + 7}{( x - 5 ) ( x - 4 )} d x

Answer: $6 x^{2} - 9 x + 20 + 34 lo g (x - \frac{9}{2}) + x^{2} - 9 x + 20 + C$

Expand

(x - 5) (x - 4) = x^{2} - 9 x + 20

. Write

6 x + 7 = 3 (2 x - 9) + 34

, where

2 x - 9 = \frac{d}{d x} (x^{2} - 9 x + 20)

. Then

\int \frac{6 x + 7}{x ^{2} - 9 x + 20} d x = 3 \int \frac{( 2 x - 9 ) d x}{x ^{2} - 9 x + 20} + 34 \int \frac{d x}{x ^{2} - 9 x + 20} .

The first integral is

3 \cdot 2 x^{2} - 9 x + 20 = 6 x^{2} - 9 x + 20

. For the second,

x^{2} - 9 x + 20 = (x - \frac{9}{2})^{2} - (\frac{1}{2})^{2}

34 lo g (x - \frac{9}{2}) + x^{2} - 9 x + 20 .

6 x^{2} - 9 x + 20 + 34 lo g (x - \frac{9}{2}) + x^{2} - 9 x + 20 + C

CBSE Class 12 Board · 2011 · Subjective