Practice Fundamental Theorem of Calculus in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

hard

Find

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

Example 2

medium

Use the FTC to evaluate

\int_{-1}^{2} (2x + 3) \, dx

Example 3

easy

Evaluate

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

Example 4

easy

G(x) = \int_3^x t^4 \, dt

, find

G'(5)

Example 5

easy

H(x) = \int_0^x e^t \, dt

, find

H'(x)

Example 6

medium

Use the FTC to evaluate

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

Example 7

easy

Evaluate

\int_0^2 3x^2 \, dx

Example 8

easy

F(x) = \int_0^x \sin t \, dt

, what is

F'(x)

Example 9

medium

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

, find

F'(x)

Example 10

easy

Use the FTC to evaluate

\int_0^3 2x \, dx

Example 11

medium

Evaluate

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

Example 12

easy

Let

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

. Find

G'(x)

using FTC Part 1.

Example 13

medium

Use FTC Part 2 to evaluate

\int_1^e \frac{1}{t}\,dt

Example 14

easy

G(x) = \int_2^x \cos t \, dt

, find

G'(x)

Example 15

hard

Why must

f

be continuous on

[a,b]

for FTC Part 2 in its basic form?

Example 16

challenge

Let

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

. Find all

x > 0

where

G

has a local minimum.

Example 17

medium

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

, find

F'(x)

Example 18

medium

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

, find

F'(x)

Example 19

easy

Use the FTC to evaluate

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

Example 20

easy

Evaluate

\int_0^1 e^x \, dx

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Related Concepts

Derivative Integral