The value of the integral `int_0^(log5)(e^xsqrt(e^x-1))/(e^x+3)dx`

Options

A. `3+2pi`
B. `4-pi`
C. `2+pi`
D. none of these

Answer

B

Explanation

Putting `e^(x)-1=t^(2)` in the given integral, we have
`int_(0)^(log5) (e^(x)sqrt(e^(x)-1))/(e^(x)+3) dx=2 int_(0)^(2) (t^(2))/(t^(2)+4) dt =2 (int_(0)^(2) 1 dt -4int_(0)^(2) (dt)/(t^(2)+4))`
`=2[(t-2 tan^(-1)(t/2))_(0)^(2)]`
`=2[(2-2xxpi//4)]=4-pi`