If ∫ e^x≤ft((x-1)/((x+1)^3))dx = (A. e^x)/((x + 1)^B)+C , where C is…

The correct answer is Option (3) → (B) and (D) only

The integrand is put in the form \(e^x[f(x) + f'(x)]\), whose integral is \(e^x f(x) + C\):

    \(\displaystyle \frac{x - 1}{(x + 1)^3} = \frac{(x + 1) - 2}{(x + 1)^3} = \frac{1}{(x + 1)^2} - \frac{2}{(x + 1)^3}\)

With \(f(x) = \frac{1}{(x + 1)^2}\):

    \(\displaystyle f'(x) = -\frac{2}{(x + 1)^3}\)

So

    \(\displaystyle \int e^x\left(\frac{x - 1}{(x + 1)^3}\right)dx = \frac{e^x}{(x + 1)^2} + C\)

So A = 1 and B = 2.

Checking the statements

(A) A = −1 — Incorrect. A is 1.

(B) A = 1 — Correct. The coefficient of \(\frac{e^x}{(x + 1)^2}\) is 1.

(C) B = 3 — Incorrect. B is 2.

(D) B = 2 — Correct. The power of \(x + 1\) is 2.