Question

Evaluate the following integrals​

Answer 1

`I = (1)/(8) e^(4x^2) + C`

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 = (1)/(8)dt`

`\implies \: I = \int e^(t) (1)/(8)dt`

`\implies \: I = (1)/(8) \int e^(t)dt`

`\implies \: I = (1)/(8) e^(t) + C`

`\implies \: I = (1)/(8) e^(4x^2) + C`

`\implies \: \int xe^(4x^2) dx = (1)/(8) e^(4x^2) + C`