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: The Fundamental Theorem links the two operations: the derivative of an accumulation function gives back the integrand, and a definite integral equals the change in any antiderivative.

Common stuck point: The procedure for fundamental theorem of calculus is the easy part; the trap is mixing up the parts. Asking "Am I connecting a definite integral to an antiderivative ( $F(b)-F(a)$ ) or differentiating an accumulation function back to its integrand?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I connecting a definite integral to an antiderivative ( $F(b)-F(a)$ ) or differentiating an accumulation function back to its integrand?

Worked Examples

Example 1

easy

Let

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

. Find

G'(x)

using FTC Part 1.

Answer

G^{'} (x) = x^{2} + 1

First step

FTC Part 1 states: if

G(x) = \int_a^x f(t)\,dt

, then

G'(x) = f(x)

Full solution

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 $x$ .

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} d t

Example 2

medium

F(x) = \int_0^x e^{t^2}\,dt

, find

F'(x)

Example 3

easy

Use the FTC to evaluate

\int_0^3 2x \, dx

Example 4

easy

medium

Use the FTC to evaluate

\int_0^2 (e^x - 1) \, dx

Example 15

medium

Distinguish: which is FTC Part 1 vs Part 2 — (a)

\frac{d}{dx}\int_a^x f\,dt = f(x)

, (b)

\int_a^b f = F(b)-F(a)

Example 16

medium

F(x) = \int_1^x \frac{1}{t}\,dt

, find

F'(x)

and

F(1)

Example 17

medium

Use the FTC to evaluate

\int_0^4 \sqrt{x}\,dx

Example 18

challenge

F(x) = \int_{x}^{x^2} t \, dt

, find

F'(x)

Example 19

challenge

A particle has velocity

v(t) = 3t^2 - 2

. Find the net displacement from

t=0

t=2

Example 20

challenge

\frac{d}{dx}\int_0^{\sin x} e^{t^2}\,dt = ?

, express the derivative.

Example 21

medium

Use the FTC to evaluate

\int_0^2 (3x^2 - 4x + 1) \, dx

Example 22

medium

easy

Evaluate

\int_0^4 \sqrt{x} \, dx

Example 29

medium

F(x) = \int_1^{x^3} \frac{1}{t} \, dt

, find

F'(x)

Example 30

medium

Evaluate

\int_0^{\pi} \sin x \, dx

Example 31

medium

F(x) = \int_{2x}^{5} (t^2 + 1) \, dt

, find

F'(x)

Example 32

medium

Evaluate

\int_1^4 \frac{2x - 1}{\sqrt{x}} \, dx

Example 33

medium

A car's velocity is

v(t) = 2t + 1

m/s. Use FTC to find total distance traveled from

t = 0

t = 4

Example 34

medium

F(x) = \int_0^x t e^{-t^2} \, dt

, find

F'(x)

Example 35

medium

Evaluate

\int_0^{\ln 2} 2 e^{2x} \, dx

Example 36

medium

State and apply: if

f

is continuous and

\int_2^x f(t) dt = x^2 - 4

, find

f(x)

Example 37

hard

Find

\frac{d}{dx} \int_{x}^{x^2} \sin t \, dt

Example 38

hard

Evaluate

\int_0^{\pi/4} \sec^2 x \, dx

Example 39

hard

A water tank's volume is

V(t) = 50 + \int_0^t (6 - r) dr

. Find

V'(t)

and the maximum volume.

Example 40

hard

Evaluate

\int_0^1 \frac{2x}{1 + x^2} \, dx

Example 41

hard

Evaluate

\int_1^2 \ln x \, dx

Example 42

hard

F(x) = \int_0^{\cos x} t^2 \, dt

, find

F'(x)

Example 43

challenge

Use FTC to find the average value of

f(x) = x^2

[0, 3]

Example 44

challenge

Let

G(x) = \int_0^x (t - 1)(t - 3) \, dt

. Find all

x > 0

where

G

has a local minimum.

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Related Concepts

Derivative Integral

Background Knowledge

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

derivativeintegral