Maths. Section B1Indefinite Integration

If ∫ f(x) d x=F(x), then ∫ x^3 f≤ft(x^2) d x is equal to

If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to

Answer & explanation

Correct answer: option 2

We have, $\int f(x) d x=F(x)$

∴ $I=\int x^3 f\left(x^2\right) d x=\frac{1}{2} \int x^2 f\left(x^2\right) d\left(x^2\right)$

$\Rightarrow I =\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$

$\Rightarrow I=\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$