Question

`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3) * dx` = ______.

Options

• 3 + 2π

• 2 + π

• 4 – π

• 4 + π

Answer 1

`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3) * dx` = 4 – π.

Explanation:

Putting e^x – 1 = t² in the given integral, we have

also `d/dx (e^x - 1) = d/dx t^2`

⇒ `e^x = 2t dt/dx`

`int_0^log 5 (e^x sqrt(e^x - 1))/(e^x + 3) dx`

= `2 int_0^2 t^2/(t^2 + 4) dt`

= `2 int_0^2 (t^2 + 4 - 4)/(t^2 + 4) dt`

= `2(int_0^2 1 dt - 4 int_0^2 dt/(t^2 + 4)`

= `2 [(t - 2 tan^(-1) (t/2))_0^2]`

= `2 [(2 - 2 xx pi/4)]`

= 4 – π