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Question: If\(\int_{}^{}\frac{2x^{2} + 3}{(x^{2} - 1)(x^{2} + 4)}\)dx = A log\(\left( \frac{x + 1}{x - 1} \rig...

If∫2x2+3(x2−1)(x2+4)\int_{}^{}\frac{2x^{2} + 3}{(x^{2} - 1)(x^{2} + 4)}dx = A log(x+1x−1)\left( \frac{x + 1}{x - 1} \right)+ B tan–1(x2)\left( \frac{x}{2} \right), then

(A, B) is-

A

(– 1/2, 1/2)

B

(1/2, –1/2)

C

(–1, 1)

D

(1, –1)

Answer

(– 1/2, 1/2)

Explanation

Solution

∫(2x2+3)dx(x2−1)(x2+4)\int \frac { \left( 2 x ^ { 2 } + 3 \right) d x } { \left( x ^ { 2 } - 1 \right) \left( x ^ { 2 } + 4 \right) } = ∫(1x2−1+1x2+4)\int_{}^{}\left( \frac{1}{x^{2} - 1} + \frac{1}{x^{2} + 4} \right)dx

= 12\frac{1}{2}log (x−1x+1)\left( \frac{x - 1}{x + 1} \right) + 12\frac{1}{2} tan–1x2\frac{x}{2} + c

̃ a = –12\frac{1}{2}, b = 12\frac{1}{2}