Fundamental Theorem of Calculus Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Fundamental Theorem of Calculus.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

Integration undoes differentiation. They're two sides of the same coin.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Part 1: Derivative of integral = original function. Part 2: \int_a^b f'(x) \, dx = f(b) - f(a).

Common stuck point: The FTC is why we can use antiderivatives to compute definite integrals.

Sense of Study hint: Write down which part of FTC applies, then check whether the upper limit is just x or a function of x.

Worked Examples

Example 1

easy
Let G(x) = \int_0^x (t^2 + 1)\,dt. Find G'(x) using FTC Part 1.

Solution

  1. 1
    FTC Part 1 states: if G(x) = \int_a^x f(t)\,dt, then G'(x) = f(x).
  2. 2
    Here f(t) = t^2 + 1, so G'(x) = f(x) = x^2 + 1.
  3. 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.

Example 2

hard
Let H(x) = \int_1^{x^2} \cos t\,dt. Find H'(x).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Use FTC Part 2 to evaluate \int_1^e \frac{1}{t}\,dt.

Example 2

medium
If F(x) = \int_0^x e^{t^2}\,dt, find F'(x).
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Related Concepts

DerivativeIntegral

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativeintegral