The value of `int_(1)^(3)(sqrt(1+(x-1)^(3))+(x^(2)-1)^(1/3)+1)dx` is __________.

Options

A. `(pi)/8 log_(e)2`
B. `(pi)/2 log_(e)2`
C. `-(pi)/2 log_(e)2`
D. none of these

Answer

A

Explanation

`int_(1)^((1+sqrt(5))/2) (1+1/(x^(2)))/(x^(2)-1+1/(x^(2)))"log"(1+x-1/x)dx`
`=int_(1)^((1+sqrt(5))/2)(1+1/(x^(2)))/((x-1/x)^(2)+1)log(1+x-1/x)dx`
Put`x-1/x=t`
or `(1+1/(x^(2)))dx=dt`
If `x=1,t=0` and if `x=(sqrt(5)+1)/2,t=1`
`:. I=int_(0)^(1)(In(1+t)dt)/(1+t^(2))`
Put `t=tan theta` and `dt =sec^(2) theta d theta`
`:. I=int_(0)^(pi//4) In (1+tan theta)d theta =(pi)/8 log_(e)2`