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If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to |
$\frac{1}{2}\left[x^2\{F(x)\}^2-\int\{F(x)\}^2 d x\right]$ $\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$ $\frac{1}{2}\left[x^2 F(x)-\frac{1}{2} \int\{F(x)\}^2 d x\right]$ none of these |
$\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$ |
We have, $\int f(x) d x=F(x)$ ∴ $I=\int x^3 f\left(x^2\right) d x=\frac{1}{2} \int x^2 f\left(x^2\right) d\left(x^2\right)$ $\Rightarrow I =\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$ $\Rightarrow I=\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$ |
