Question

Evaluate the following integrals​

Answer 1

Answer:

$I = \frac{1}{8} e^{4x^2} + C$

Step-by-step explanation:

This can be solved using substitution method. Let us work it out.

$Let \: I = \int xe^{4x^2} dx$

$\implies \: I = \int e^{4x^2} xdx$

$Put \:4x^2 = t$

$\implies 8x dx = dt$

$\implies x dx = \frac{1}{8}dt$

$\implies \: I = \int e^{t} \frac{1}{8}dt$

$\implies \: I = \frac{1}{8} \int e^{t}dt$

$\implies \: I = \frac{1}{8} e^{t} + C$

$\implies \: I = \frac{1}{8} e^{4x^2} + C$

$\implies \: \int xe^{4x^2} dx = \frac{1}{8} e^{4x^2} + C$