Question

Integrate the following with respect to x.

e^x(1 + x) log(xe^x)

Answer 1

e^x(1 + x) log(xe^x) = (e^x + xe^x) log(xe^x)

Let z = xe^x

Then dz = d(xe^x)

dz = (xe^x + e^x) dx  .......(Using product rule)

So `int "e"^x (1 + x) log(x"e"^x)  "d"x`

= `int log (x"e"^x) ("e"^x + x"e"^x)  "d"x`

= `int log z  "d"z`

= z(log z – 1) + c

= xe^x [log (xe^x) – 1] + c