Question

If `int f(x)dx=F(x),` then `intx^3f(x^2)dx` is equal to :
A. `(1)/(2)[x^(2){F(x)}^(2)-int{F(x)}^(2)dx]`
B. `(1)/(2)[x^(2)F(x^(2))-intF(x^(2))d(x^(2))]`
C. `(1)/(2)[x^(2)F(x)-(1)/(2)int{F(x)}^(2)dx]`
D. none of these

Answer 1

Correct Answer - b
We have , `intf(x)dx=F(x)`.
`thereforeI=intx^(3)f(x^(2))dx=(1)/(2)intunderset(I)(x^(2))f(x^(2))underset(II" ")(d(x^(2)))`
`rArrI=(1)/(2)[x^(2)F(x^(2))=intF(x^(2))d(x^(2))]`
`rArrI=(1)/(2)[x^(2)F(x^(2))-intF(x^(2))d(x^(2))]`