Evaluate the following integrals: ∫ ((x + 1)/(x + 2)^2) e^x dx

Question

Evaluate the following integrals:

\(\int \frac{(x + 1)}{(x + 2)^2}e^x dx\)

∫ ((x + 1)/(x + 2)²) e^x dx

Answer 1

 

((x + 1)/(x + 2)²)

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

where c is the integrating constant.