Evaluate `int_(2)^(3)(x^2+1)dx`

Answer

C

Explanation

Let `F(k)=int_(0)^(pi//2) In (sin^(2)theta+k^(2)cos^(2)theta) d theta`
`F'(k)=int_(0)^(pi//2)1/(sin^(2)theta+k^(2)cos^(2)theta) 2kcos^(2) theta d theta`
`=2kint_(0)^(pi//2) (cos^(2)theta)/(sin^(2) theta+k^(2)cos^(2)theta) d theta`
`=2k int_(0)^(pi//2) (d theta)/(tan^(2) theta +k^(2))`
`=2k int_(0)^(pi//2) (sec^(2)theta-tan^(2)theta)/(tan^(2) theta+k^(2)) d theta`
`=2kint_(0)^(oo) (dt)/(t^(2)+k^(2))-2kint_(0)^(pi//2) d theta +2k^(3) int_(0)^(pi//2) (d theta)/(tan^(2) theta +k^(2))`
`=2k 1/k "tan"^(-1) t/k|_(0)^(oo) -2k (pi)/2 +k^(2)F'(k)`
or `(1-k^(2))F'(k)=pi-kpi=pi(1-k)`
or `F'(k)=(pi)/(1+k)`
or `F(k)=pi log (1+k)+c`
For `k=1, F(1)=0` or `c=-pilog2`
or `F(k)=pilog(1+k)-pilog2`