Question

`int (x + 1)/(x(1 + xe^x)) dx` is equal to

Options

• `log|(1 + xe^x)/(xe^x)| + C`

• `log|(xe^x)/(1 + xe^x)| + C`

• log |xe^x (1 + xe^x)| + C

• log |1 + xe^x| + C

Answer 1

`bb(log|(xe^x)/(1 + xe^x)| + C)`

Explanation:

Let `I = int (x + 1)/(x(1 + xe^x)) dx`

Put `xe^x = t = (x + 1)dx = (dt)/(e^x)`

= `I = int (dt)/(xe^x (1 + xe^x)) = int 1/(t(1 + t))dt = int (t + 1 - t)/(t(t + 1))dt`

∴ `I = int (1/t - 1/(t + 1))dt`

= `log t - log (t + 1) + C = log|t/(t + 1)| + C`

= `log|(xe^x)/(xe^x + 1)| + C`