If `int_(0)^(x)((t^(3)+t)dt)/((1+3t^(2)))=f(x)`, then `int_(0)^(1)f^(')(x) dx` is equal to

Options

A. `1/6-1/9 In 3`
B. `1/6-1/9 In 4`
C. `1/6+1/4 In 3`
D. `1/6+1/9 In 4`

Answer

D

Explanation

`:. int_(0)^(x) (t^(3))/(1+3t^(2)) dt + int_(0)^(x) (tdt)/(1+3t^(2))=f(x)`
Put `1+3t^(2)=u^(2)implies`
`=int (1/3 udu xx((u^(2)-1)/3))/(u^(2))+1/6 In (1+3t^(2))|_(0)^(x)`
`=1/9 int_(1)^(sqrt(1+3x^(2)) (u-1/u) du+1/6 In (1+3x^(2))`
`=1/9 ((u^(2))/2-Inu)_(1)^(sqrt(1+3x^(2))+1/6 In (1+3x^(2))`
`=1/18 ((1+3x^(2))-1)-1/9 In sqrt(1+3x^(2))+ 1/6 In (1+3x^(2))`
`=1/6 x^(2)+((-1)/18+1/6)` In `(1+3x^(2))`
`=1/6 x^(2)-1/9 In (1+3x^(2))`