Question

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

Options

• 3 + 2π

• 4 - π

• 2 + π

• 4 + π

Answer 1

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

Explanation:

Put e^x - 1 = t² ⇒ e^x dx = 2t dt

∴ `int_0^log5 (e^xsqrt(e^x - 1))/(e^x + 3)` dx = `int_0^2 (2t^2dt)/(t^2 + 4) = 2int_0^2(1 - 4/(t^2 + 4))dt`

= `2[t - 4(1/2tan^-1  t/2)]_0^2`

= 4 - π