Mathematics - Definite Integration Question with Solution | TestHub

To refine the solution, we apply the Fundamental Theorem of Calculus.

Given:

$${\int }_0^{x^3}{t}^{2}f(t)dt=\frac{3}{13}{x}^{13}+5$$

Differentiate both sides with respect to $x$ using the Leibniz integral rule:

$$(x^3{)}^{2}f(x^3)\cdot \frac{d}{dx}(x^3)=\frac{d}{dx}\left(\frac{3}{13}{x}^{13}+5\right)$$

$$x^{6}f(x^3)\cdot 3x^2=3x^{12}$$

$$3x^{8}f(x^3)=3x^{12}$$

For $x\ne 0$:

$$f(x^3)=x^4$$

We need to find $f\left(\frac{8}{27}\right)$.

Let $x^3=\frac{8}{27}$. Then $x=\sqrt[3]{\frac{8}{27}}=\frac{2}{3}$.

Substitute $x=\frac{2}{3}$ into $f(x^3)=x^4$:

$$f\left(\frac{8}{27}\right)={\left(\frac{2}{3}\right)}^4$$

$$f\left(\frac{8}{27}\right)=\frac{2^4}{3^4}=\frac{16}{81}$$

The final answer is $\boxed{\frac{16}{81}}$.