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If int f(x)dx=psi(x), then int x^5f(x^3)...

If `int f(x)dx=psi(x)`, then `int x^5f(x^3)dx`

A

`(1)/(3)[x^(3)psi(x^(3))-int x^(2)psi(x^(3))dx]+c`

B

`(1)/(3)x^(3)psi(x^(3))-3 int x^(3)psi(x^(3))dx+c`

C

`(1)/(3)x^(3)psi(x^(3))-int x^(2)psi(x^(3))dx+c`

D

`(1)/(3)[x^(3)psi(x^(3))-int x^(3)psi(x^(3))dx]+c`

Text Solution

Verified by Experts

The correct Answer is:
C
Given, `int f(x)dx = psi(x)`
Let `" "I=intx^(5)f(x^(3))dx`
Put `" "x^(3)=t`
`rArr" "x^(2)dx=(dt)/(3)" "...(i)`
`therefore I=(1)/(3)int t f(t) dt`
`=(1)/(3)[t*intf(t)dt-int{(d)/(dt)(t)intf(t)dt}dt]" "["integration by parts"]`
`=(1)/(3)[t psi (t)-int psi (t)dt]`
`=(1)/(3)[x^(3) psi (x^(3))-3 int x^(2) psi(x^(3))dx]+c" "["from Eq. (i)"]`
`=(1)/(3)x^(3) psi (x^(3))-int x^(2) psi (x^(3))dx+c`
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