`int e^(x^4) (x + x^3 +2x^5) e^(x^2) dx` is equal to

Options

A. `(1)/(2)xe^(x^(2))e^(x^(4))+C`
B. `(1)/(2)x^(2)e^(x^(4))+C`
C. `(1)/(2)e^(x^(2))e^(x^(4))+C`
D. `(1)/(2)x^(2)e^(x^(2))e^(x^(4))+C`

Answer

d

Explanation

Let `x^(2)=t` then ,
`I=e^(x^(4))(x+x^(3)+2x^(5))e^(x^(2))dx=(1)/(2)inte^(t^(2))(1+t+2t^(2))e^(t)dt`
`rArrI=(1)/(2)inte^(t){underset(f)(te^(t^(2)))+underset(f)((e^(t^(2))+2t^(2)e^(t^(2))))}dt`
`rArrI=(1)/(2)e^(t)(te^(t^(2)))+C=(1)/(2)x^(2)ex^(2)e^(x^(4))+C`