Question

Combining derivatives and integrals: Simplify each of the following as much as possible.

$\frac{d}{dx}{\int }_0^x t^{3}dt$

Short answer

$\frac{d}{dx}{\int }_0^x t^{3}dt=x^3$

Step-by-step solution

Step 1. Given information.

Given expression is$\frac{d}{dx}{\int }_0^x t^{3}dt$

We have to simply solve the expression.

Step 2. Solve the expression.

If f is continuous on [a,b] then for all $x\in [a,b]$

$\frac{d}{dx}{\int }_a^x f\left(t\right)dt=f\left(x\right)$

So

$\begin{array}{r}f\left(t\right)=t^3\\ f\left(x\right)=x^3\end{array}$

And

$\begin{array}{r}\frac{d}{dx}{\int }_0^x t^{3}dt\\ =x^3\end{array}$