how to do this integration?

Question

integration [x*(e^2x)*dx]

Answer 1

the integral of [x(e^2x)dx] is:

(1/2)xe^2x - (1/4)e^2x + c

b/c the integral of x(e^ax)dx is:

(1/a)xe^ax - (1/2a)e^ax + c

Answer 2

You have to do this integration by parts.

Let u = x. dv = e^(2x) dx
du = dx. v = (1/2)e^(2x)

When doing integration by parts, it follows this formula:

uv - Integral (v du)

Substituting directly,

x(1/2)e^(2x) - Integral ( (1/2)e^(2x) dx)

Pulling out the constant 1/2, we get

x(1/2)e^(2x) - (1/2) Integral (e^(2x) dx)

Now, the integration is simple.

x(1/2)e^(2x) - (1/2) [ (1/2) e^(2x) ] + C

(1/2)xe^(2x) - (1/4) e^(2x) + C

Answer 3

integrating by parts

= x*1/2 e^2x -integral (1/2e^2x) = 1/2e^2x*(x-1/2)