Definite Integration

On this placeholder page we present the essentials of definite integration, including how to use the power rule, exponential rule, and trigonometric rules. Of course we have typical problems below, each with a complete solution immediately available, so you can see how they are used routinely, including in some typical exam questions. Let's jump right in!

Here are the main rules you'll need. We have illustration of each problem type below.

MATHENO ESSENTIALS: Definite Integration

Important Notation

When we write $[𝐹 (𝑥)] 𝑏 𝑎$ , it means $𝐹 (𝑏)$ minus $𝐹 (𝑎)$ :

[𝐹 (𝑥)] 𝑏 𝑎 = 𝐹 (𝑏) - 𝐹 (𝑎)

Power Rule: Integration of $𝑥 𝑛$ $(𝑛 \neq - 1)$

If you're integrating x-to-some-power (except $𝑥 - 1$ ), the rule to remember is: "Increase the power by 1, and then divide by the new power." We can express this process mathematically as $\int 𝑏 𝑎 𝑥 𝑛 𝑑 𝑥 = 1 𝑛 + 1 [𝑥 𝑛 + 1] 𝑏 𝑎 (𝑛 \neq - 1) = 1 𝑛 + 1 [𝑏 𝑛 + 1 - 𝑎 𝑛 + 1]$

Exponential Rule: Integration of $𝑒 𝑥$

This integral is the easiest to remember: since $(𝑒 𝑥)' = 𝑒 𝑥$ , the integral of $𝑒 𝑥$ is also $𝑒 𝑥$ $\int 𝑏 𝑎 𝑒 𝑥 𝑑 𝑥 = [𝑒 𝑥] 𝑏 𝑎 = 𝑒 𝑏 - 𝑒 𝑎$

Exponential Rule: Integration of $𝑐 𝑥$ (where $𝑐$ is a constant)

$\int 𝑏 𝑎 𝑐 𝑥 = 1 l n 𝑐 [𝑐 𝑥] 𝑏 𝑎 = 1 l n 𝑐 [𝑐 𝑏 - 𝑐 𝑎]$

Trigonometric Rules: Integrals of Trig Functions

$\int 𝑏 𝑎 c o s 𝑥 𝑑 𝑥 = [s i n 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 s i n 𝑥 𝑑 𝑥 = - [c o s 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 s e c 2 𝑥 𝑑 𝑥 = [t a n 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 s e c 𝑥 t a n 𝑥 𝑑 𝑥 = [s e c 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 c s c 2 𝑥 𝑑 𝑥 = - [c o t 𝑥] 𝑏 𝑎$

You might also be expected to know the integrals for $s i n 2 𝑥$ and $c o s 2 𝑥$ , because they follow immediately from use of the half-angle formulas:

$\int 𝑏 𝑎 s i n 2 𝑥 𝑑 𝑥 = \int 𝑏 𝑎 (12 - 12 c o s 2 𝑥) 𝑑 𝑥 = [12 𝑥 - 14 s i n 2 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 c o s 2 𝑥 𝑑 𝑥 = \int 𝑏 𝑎 (12 + 12 c o s 2 𝑥) 𝑑 𝑥 = [12 𝑥 + 14 s i n 2 𝑥] 𝑏 𝑎$

You can practice each of these common integration problems below.

I. Power Rule

If you're integrating x-to-some-power (except $𝑥 - 1$ ), the rule to remember is:

"Increase the power by 1, and then divide by the new power."

We can express this process mathematically as

\int 𝑏 𝑎 𝑥 𝑛 𝑑 𝑥 = 1 𝑛 + 1 [𝑥 𝑛 + 1] 𝑏 𝑎 (𝑛 \neq - 1) = 1 𝑛 + 1 [𝑏 𝑛 + 1 - 𝑎 𝑛 + 1]

For example

Power Rule Practice Problem #1

(a)

Evaluate the integral:

\int 51 7 𝑑 𝑥 .

(b)

Evaluate the integral:

\int 51 𝑥 𝑑 𝑥 .

(c)

Evaluate the integral:

\int 51 (7 + 𝑥) 𝑑 𝑥 .

(a) 28
(b) 12
(c) 40

[close solution]

Power Rule Practice Problem #2

(a)

Evaluate the integral:

\int 20 𝑥 2 𝑑 𝑥 .

(b)

Evaluate the integral:

\int 20 5 𝑥 4 𝑑 𝑥 .

(c)

Evaluate the integral:

\int 20 (𝑥 2 - 5 𝑥 4) 𝑑 𝑥 .

(a) $83$
(b) 32
(c) $- 2913$

[close solution]

Power Rule Practice Problem #3

Evaluate the integral:

\int 0 - 3 (23 𝑤 5 - 14 𝑤 3 + 𝑤) 𝑑 𝑤 .

Power Rule Practice Problem #4

Evaluate the integral:

\int 05 (𝑥 2 - 1) 2 𝑑 𝑥 .

Power Rule Practice Problem #5

Evaluate the integral:

\int 21 4 𝑥 - 7 𝑥 5 𝑥 3 𝑑 𝑥 .

Power Rule Practice Problem #6

(a)

Evaluate the integral:

\int 91 \sqrt 𝑥 𝑑 𝑥 .

(b)

Evaluate the integral:

\int 91 1 \sqrt 𝑥 𝑑 𝑥 .

(c)

Evaluate the integral:

\int 91 (\sqrt 𝑥 - 1 \sqrt 𝑥) 𝑑 𝑥 .

(a) $523$
(b) 4
(c) $403$

[close solution]

Power Rule Practice Problem #7

Evaluate the integral:

\int 94 𝑥 - 5 \sqrt 𝑥 𝑑 𝑥 .

Power Rule Practice Problem #8

Evaluate the integral:

\int 10 (5 \sqrt 𝑥 5 + 2 3 \sqrt 𝑥 2) 𝑑 𝑥 .

II. Exponential, $𝑒 𝑥$

This integral is the easiest to remember: since $(𝑒 𝑥)' = 𝑒 𝑥$ , the integral of $𝑒 𝑥$ is also $𝑒 𝑥 :$

\int 𝑏 𝑎 𝑒 𝑥 𝑑 𝑥 = [𝑒 𝑥] 𝑏 𝑎 = 𝑒 𝑏 - 𝑒 𝑎

Exponential Rule

𝑒 𝑥

Practice Problem #1

Evaluate the integral:

\int 31 5 𝑒 𝑥 𝑑 𝑥 .

Exponential Rule

𝑒 𝑥

Practice Problem #2

Evaluate the integral:

\int 41 (1 \sqrt 𝑥 5 - 𝑒 𝑥) 𝑑 𝑥 .

III. Exponential, $𝑐 𝑥$ (where $𝑐$ is a constant)

Here $𝑐$ is a constant (a number, any number). Then

\int 𝑏 𝑎 𝑐 𝑥 = 1 l n 𝑐 [𝑐 𝑥] 𝑏 𝑎 = 1 l n 𝑐 [𝑐 𝑏 - 𝑐 𝑎]

Exponential Rule

𝑐 𝑥

Practice Problem #1

Evaluate the integral:

\int 20 3 𝑥 𝑑 𝑥 .

Exponential Rule

𝑐 𝑥

Practice Problem #2

Evaluate the integral:

\int 10 (𝑥 5 + 5 𝑥) 𝑑 𝑥 .

IV. Trig Functions

We're listing here only the trig integrals that you should be familiar with at this early stage: each of these follows directly from a derivative you immediately know. For example, since
$(s i n 𝑥)' = c o s 𝑥$
we know immediately that
$\int 𝑏 𝑎 c o s 𝑥 𝑑 𝑥 = [s i n 𝑥] 𝑏 𝑎$
Accordingly:

$\int 𝑏 𝑎 c o s 𝑥 𝑑 𝑥 = [s i n 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 s i n 𝑥 𝑑 𝑥 = - [c o s 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 s e c 2 𝑥 𝑑 𝑥 = [t a n 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 s e c 𝑥 t a n 𝑥 𝑑 𝑥 = [s e c 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 c s c 2 𝑥 𝑑 𝑥 = - [c o t 𝑥] 𝑏 𝑎$
Each integral follows directly from a derivative you know. You can review those using our Trig Function Derivatives table; it's always available from the Reference menu at the top of every page.

You might also be expected to know the integrals for $s i n 2 𝑥$ and $c o s 2 𝑥$ , because they follow immediately from use of the half-angle formulas:

s i n 2 𝑥 = 12 - 12 c o s 2 𝑥 c o s 2 𝑥 = 12 + 12 c o s 2 𝑥

Then

\int 𝑏 𝑎 s i n 2 𝑥 𝑑 𝑥 = \int (12 - 12 c o s 2 𝑥) 𝑑 𝑥 = [12 𝑥 - 14 s i n 2 𝑥] 𝑏 𝑎 \int 𝑏 𝑎 c o s 2 𝑥 𝑑 𝑥 = \int (12 + 12 c o s 2 𝑥) 𝑑 𝑥 = [12 𝑥 + 14 s i n 2 𝑥] 𝑏 𝑎

Trig Function Practice Problem #1

Evaluate the integral:

\int 𝜋 / 2 0 4 c o s 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #2

Evaluate the integral:

\int 𝜋 0 12 s i n 𝜃 𝑑 𝜃 .

Trig Function Practice Problem #3

Evaluate the integral:

\int 𝜋 / 4 0 𝑑 𝑥 s e c 𝑥 .

Trig Function Practice Problem #4

Evaluate the integral:

\int 𝜋 / 4 0 s e c 2 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #5

Evaluate the integral:

\int 𝜋 / 3 𝜋 / 6 s e c 𝑥 t a n 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #6

Evaluate the integral:

\int 3 𝜋 / 4 𝜋 / 4 (c s c 2 𝑥 + 2 𝑥 2) 𝑑 𝑥 .

Trig Function Practice Problem #7

Evaluate the integral:

\int 𝜋 / 3 0 3 + 2 c o s 2 𝑥 c o s 2 𝑥 𝑑 𝑥 .

You may also be expected to use the Trig Identity and its variants:

s i n 2 𝑥 + c o s 2 𝑥 = 1 1 + c o t 2 𝑥 = c s c 2 𝑥 1 + t a n 2 𝑥 = s e c 2 𝑥

We'll of course illustrate the use of these identities in the problems below.

Trig Function Practice Problem #8

Evaluate the integral:

\int 𝜋 / 4 0 t a n 2 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #9

Evaluate the integral:

\int 𝜋 / 4 0 5 s i n 𝑥 + 5 s i n 𝑥 t a n 2 𝑥 s e c 2 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #10

Evaluate the integral:

\int 𝜋 / 3 𝜋 / 6 s e c 3 𝑥 - s e c 𝑥 t a n 𝑥 𝑑 𝑥 .

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Definite Integration

Important Notation

Power Rule: Integration of 𝑥𝑛 (𝑛 ≠ −1)

Exponential Rule: Integration of 𝑒𝑥

Exponential Rule: Integration of 𝑐𝑥 (where 𝑐 is a constant)

Trigonometric Rules: Integrals of Trig Functions

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Power Rule: Integration of $𝑥 𝑛$ $(𝑛 \neq - 1)$

Exponential Rule: Integration of $𝑒 𝑥$

Exponential Rule: Integration of $𝑐 𝑥$ (where $𝑐$ is a constant)