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

Question

Evaluate the following integrals:

\(\int e^x \left(tan^{-1}x + \frac{1}{1 + x^2}\right)dx\)

∫ e^x (tan^-1x + 1/(1 + x²)) dx

Answer 1

 \(\int e^x \left(tan^{-1}x + \frac{1}{1 + x^2}\right)dx\)

∫ e^x (tan^-1x + 1/(1 + x²)) dx

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) = tan^-1x and f₂(x) = e^x in the first integral and keeping the second integral intact,

where c is the integrating constant