CBSE Class 12 · 2024 · Indefinite IntegralsShort Answer

Find

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

Answer: $- \frac{3}{4} lo g ∣ x + 1∣ - \frac{1}{2 ( x + 1 )} + \frac{3}{4} lo g ∣ x - 1∣ + C$

Solution

Resolve into partial fractions:

\frac{2 x + 1}{( x + 1 ) ^{2} ( x - 1 )} = \frac{A}{x + 1} + \frac{B}{( x + 1 ) ^{2}} + \frac{D}{x - 1} .

Clearing denominators and comparing gives

A = - \frac{3}{4}, B = \frac{1}{2}, D = \frac{3}{4}

. Hence

\int [- \frac{3}{4 ( x + 1 )} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{3}{4 ( x - 1 )}] d x .

Integrating,

- \frac{3}{4} lo g ∣ x + 1∣ - \frac{1}{2 ( x + 1 )} + \frac{3}{4} lo g ∣ x - 1∣ + C

CBSE Class 12 Board · 2024 · Subjective