Fundamental Theorem of Calculus Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy

Let

G(x) = \int_0^x (t^2 + 1)\,dt

. Find

G'(x)

using FTC Part 1.

Solution

1
FTC Part 1 states: if $G(x) = \int_a^x f(t)\,dt$ , then $G'(x) = f(x)$ .
2
Here $f(t) = t^2 + 1$ , so $G'(x) = f(x) = x^2 + 1$ .
3
No integration is needed — the derivative of an integral with variable upper limit is just the integrand evaluated at $x$ .

Answer

G'(x) = x^2 + 1

FTC Part 1 says differentiation undoes integration when the upper limit is the variable. You simply replace

t

with

x

in the integrand. This is the key link showing derivatives and integrals are inverse operations.

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].

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