Chapter 4: Q. 40 (page 405)

Combining derivatives and integrals: Simplify each of the following as much as possible.
$\frac{d}{d x} \int_{0}^{x} t^{3} d t$

Short Answer

Expert verified

$\frac{d}{d x} \int_{0}^{x} t^{3} d t = x^{3}$

Step by step solution

Step 1. Given information.

Given expression is $\frac{d}{d x} \int_{0}^{x} t^{3} d t$

We have to simply solve the expression.

Step 2. Solve the expression.

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

$\frac{d}{d x} \int_{a}^{x} f (t) d t = f (x)$

$\begin{aligned} f (t) = t^{3} \\ f (x) = x^{3} \end{aligned}$

And

$\begin{aligned} \frac{d}{d x} \int_{0}^{x} t^{3} d t \\ = x^{3} \end{aligned}$

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Combining derivatives and integrals: Simplify each of the following as much as possible.
$\frac{d}{d x} \int_{0}^{x} t^{3} d t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the expression.

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Combining derivatives and integrals: Simplify each of the following as much as possible.ddx∫0x t3dt

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the expression.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Combining derivatives and integrals: Simplify each of the following as much as possible.
$\frac{d}{d x} \int_{0}^{x} t^{3} d t$