Chapter 4: Q. 37 (page 399)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.
$\frac{d}{d x} \int_{0}^{x^{2}} \cos t . d t$

Short Answer

Expert verified

The derivative expression of $\frac{d}{d x} \int_{0}^{x^{2}} \cos t . d t$ is $\cos x^{2} (2 x)$ .

Step by step solution

Step 1. Given Information.

The derivative:

$\frac{d}{d x} \int_{0}^{x^{2}} \cos t . d t$

Step 2. Second Fundamental theorem of calculus.

f is continuous on [a,b]for all $x \in [a, b]$ , then $\frac{d}{d x} \int_{e}^{u (x)} f (t) . d t = f (u (x)) u' (x)$

Step 3. Find the derivative expression.

By Second Fundamental theorem of calculus,

$\frac{d}{d x} \int_{0}^{x^{2}} \cos t . d t = \cos x^{2} (x^{2})' = \cos x^{2} (2 x)$

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Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.
$\frac{d}{d x} \int_{0}^{x^{2}} \cos t . d t$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Second Fundamental theorem of calculus.

Step 3. Find the derivative expression.

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Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.ddx∫0x2cost.dt

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Second Fundamental theorem of calculus.

Step 3. Find the derivative 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.

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.
$\frac{d}{d x} \int_{0}^{x^{2}} \cos t . d t$