The value of $f(\mathrm{l})$, given the equation $\int_0^{x^2} x f(t) d t=x^5-x^3$ is

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Aicharya · Accepted Answer

$\int_0^{x^2} x \cdot f(t) d t=x^5-x^3$ Differentiate w.r.t. $x$ on both sides, $ \begin{aligned} x^2 \cdot f\left(x^2ight) \cdot \frac{d}{d x} x^2-0 & =5 x^4-3 x^2 \ x^2 f\left(x^2ight)(2 x) & =5 x^4-3 x^2 \end{aligned} $ $ 2 x^3 f\left(x^2ight)=5 x^4-3 x^2 $ Put $x=1$ $ 2 \cdot 1 \cdot f(1)=5(1)-3(1) \quad \Rightarrow f(1)=1 $ Hence, option (4) is correct

The value of $f(\mathrm{l})$, given the equation $\int_0^{x^2} x f(t) d t=x^5-x^3$ is

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