`int ((x^(2)+2)(a^((x+tan^(-1)x)))/(x^(2)+1))` dx is equal to

Options

A. `log a.a^(x+tan^(-1)x)+c`
B. `((x+tan^(-1))/(log a))/(log a)+c`
C. `(a^(x+tan^(-1)x))/(log a)+c`
D. `log a(x+tan^(-1)+c`

Answer

C

Explanation

Let `I= int ((x^(2)+a^((x+tan^(-1)x)))/(x^(2)+1))dx`
Put `x+tan^(-1)x=t`
`Rightarrow (1+(1)/(1+x^(2))`dx=dt
`Rightarrow (2+x^(2))/(1+x^(2))dx=dt`
`therefore l=int a^t dt=(a^(t))/(log a)+c`
`=(a^(x+tan^(-1))/(log a)+c)`