` "If " int f(x)dx=Psi(x), " then " int x^(5)f(x)^(3)dx " is equal to "`

Options

A. `(1)/(3)x^(3){x^(3)phi(x^(3))-intx^(2)phi(x^(3))dx}+C`
B. `(1)/(3)x^(3)phi(x^(3))-3intx^(3)phi(x^(3))dx+C`
C. `(1)/(3)x^(3)phi(x^(3))-intx^(2)phi(x^(3))dx+C`
D. `(1)/(3){x^(3)phi(x^(3))-intx^(3)phi(x^(3))dx}+C`

Answer

c

Explanation

Let `x^(3)=t`. then , `d(x^(3))=3dtrArr3x^(2)dx=dt`
`thereforeintx^(5)f(x^(3))dx`
`=(1)/(3)intx^(3)f(x^(3))(3x^(2))dx`
`=(1)/(3)intunderset(I)(t)f(t)underset(II)dt`
`=(1)/(3){tphi(t)-intphi(t)dt}+C` " " `[becauseintf(t)dt=phi(t)]`
`=(1)/(3){x^(3)phi(x^(3))-3intphi(x^(3))x^(2)dx}+C`
`=(1)/(3)x^(3)phi(x^(3))-intx^(2)phi(x^(3))dx+C`