Fundamental Theorem of Calculus Math Example 2

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Example 2

hard

Let

H(x) = \int_1^{x^2} \cos t\,dt

. Find

H'(x)

.

Solution

1
The upper limit is $x^2$ , not $x$ itself, so we need the chain rule along with FTC Part 1.
2
Let $u = x^2$ , so $H(x) = \int_1^u \cos t\,dt$ .
3
By FTC Part 1, $\frac{dH}{du} = \cos u = \cos(x^2)$ .
4
By the chain rule, $H'(x) = \frac{dH}{du} \cdot \frac{du}{dx} = \cos(x^2) \cdot 2x$ .

Answer

H'(x) = 2x\cos(x^2)

When the upper limit is a function of

x

, FTC Part 1 must be combined with the chain rule. Evaluate the integrand at the upper limit, then multiply by the derivative of that upper limit.

About Fundamental Theorem of Calculus

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

Learn more about Fundamental Theorem of Calculus →

More Fundamental Theorem of Calculus Examples

Let [formula]. Find [formula] using FTC Part 1.

Example 3 medium

Use FTC Part 2 to evaluate [formula].

Example 4 medium

If [formula], find [formula].

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Related Concepts

Derivative Integral