Evaluate the following integrals: ∫ e^x (x - 1)/(x + 1)^3 dx

Question

Evaluate the following integrals:

\(\int e^x \frac{(x - 1)}{(x + 1)^3} dx\)

∫ e^x (x - 1)/(x + 1)³ dx

Answer 1

For x = -1, equation: -2 = C i.e. C = -2

For x = 0, equation: -1 = A + B - 2 i.e. A + B = 1

For x = 1, equation: 0 = 4A + 2B - 2

i.e. 2(A + B + A) = 2

⇒ 1+A = 1

⇒ A = 0

And, B = 1

Tip – If f₁(x) and f₂(x) are two functions , then an integral of the form ∫ f₁(x)f₂(x)dx can be INTEGRATED BY PARTS as

where f₁(x) and f₂(x) are the first and second functions respectively.

Taking f₁(x) = 1/(1 + x)² and f₂(x) = e^x in the first integral and keeping the second integral intact,

where c is the integrating constant.