Integral Examples: 47 Problems with Answers

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

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 reverse operation of differentiation; it also computes the exact area under a curve between two points.

If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

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: An integral is the antiderivative — the function whose derivative is the integrand — and it also accumulates a total from a rate.

Common stuck point: The procedure for integral is the easy part; the trap is forgetting the $+C$ on an indefinite integral. Asking "Am I looking for a function whose derivative is the given one (with a $+C$ ), rather than a numeric area?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I looking for a function whose derivative is the given one (with a $+C$ ), rather than a numeric area?

Worked Examples

Example 1

easy

Find

\int (4 x^{3} + 6 x) d x

Answer

x^{4} + 3 x^{2} + C

First step

Apply the power rule for integration:

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Full solution

2
For $4x^3$ : $\frac{4x^4}{4} = x^4$ .
3
For $6x$ : $\frac{6x^2}{2} = 3x^2$ .
4
Combine with the constant of integration: $x^4 + 3x^2 + C$ .

Integration reverses differentiation. The power rule for integration adds 1 to the exponent and divides by the new exponent. Always include the constant

C

for indefinite integrals.

Example 2

medium

Evaluate

\int_{0}^{2} (3 x^{2} + 1) d x

Example 3

easy

Evaluate

\int_0^3 2x \, dx

Example 4

medium

Evaluate

\int_1^4 \sqrt{x} \, dx

Example 5

medium

Find a function

f

with

f'(x) = 6x^2 - 2

and

f(0) = 5

Example 6

hard

Evaluate

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

Example 7

hard

Find the area between

y = x^2

and

y = 2x

for

x \in [0, 2]

Practice Problems

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

Example 1

easy

Find

\int (5 x^{2} - 3 x + 7) d x

Example 2

hard

Find

\int \frac{1}{x} d x

and explain why the standard power rule does not apply.

Example 3

easy

Find

\int x^2 \, dx

Example 4

easy

Find

\int 3 \, dx

Example 5

easy

Find

\int x^3 \, dx

Example 6

easy

Find

\int (2x + 1) \, dx

Example 7

easy

Find

\int e^x \, dx

Example 8

easy

Find

\int \cos x \, dx

Example 9

easy

Find

\int \frac{1}{x} \, dx

Example 10

easy

Find

\int 4x^3 \, dx

Example 11

medium

Find

\int (x^2 - 4x + 5) \, dx

Example 12

medium

Find

\int (3x^2 + 2e^x) \, dx

Example 13

medium

Find

\int \sqrt{x} \, dx

Example 14

medium

Find

\int \frac{1}{x^2} \, dx

Example 15

medium

Find

\int \sin x \, dx

Example 16

medium

Find

\int (6x^2 - \frac{2}{x}) \, dx

Example 17

medium

Verify that

F(x) = x^2 e^x

is an antiderivative pattern: find

\int (x^2 + 2x)e^x\,dx

Example 18

challenge

Find

\int (x+1)^2 \, dx

by expanding first.

Example 19

challenge

Find

\int (e^x + \cos x - \frac{3}{x}) \, dx

Example 20

challenge

Find a function

f

with

f'(x) = 4x^3 - 6x

and

f(1) = 0

Example 21

medium

Find

\int (4x^3 - \sin x) \, dx

Example 22

medium

Find

\int (2e^x + 3\cos x) \, dx

Example 23

easy

Find

\int 5 \, dx

Example 24

easy

Find

\int x^5 \, dx

Example 25

easy

Find

\int (3x^2 + 4) \, dx

Example 26

easy

Find

\int 2e^x \, dx

Example 27

medium

Find

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

Example 28

medium

Find

\int x^{-3} \, dx

Example 29

medium

Evaluate

\int_0^{\pi/2} \cos x \, dx

Example 30

medium

Find

\int (4\sin x + 3\cos x) \, dx

Example 31

medium

Evaluate

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

Example 32

medium

Find

\int \dfrac{1}{x^3} \, dx

Example 33

medium

Find

\int \dfrac{x^2 + 1}{x} \, dx

Example 34

hard

Evaluate

\int_1^e \dfrac{1}{x} \, dx

Example 35

hard

Find

\int (2x-3)^4 \, dx

Example 36

hard

Find the area under

y = x^2

from

x = 0

x = 3

Example 37

hard

A particle has velocity

v(t) = 3t^2 - 6t

m/s. Find its displacement from

t=0

t=4

Example 38

hard

Find

\int xe^{x^2} \, dx

Example 39

challenge

Find

\int \dfrac{2x}{x^2+1} \, dx

Example 40

challenge

Find the average value of

f(x) = x^2

[0, 3]

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Related Concepts

Derivative Definite Integral Fundamental Theorem of Calculus

Background Knowledge

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

derivative