Indefinite Integration

On this placeholder page we present the essentials of indefinite 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: Indefinite Integration

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 𝑥 𝑛 𝑑 𝑥 = 1 𝑛 + 1 𝑥 𝑛 + 1 + 𝐶 (𝑛 \neq - 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 𝑒 𝑥 𝑑 𝑥 = 𝑒 𝑥 + 𝐶

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

\int 𝑘 𝑥 = 1 l n 𝑘 𝑘 𝑥 + 𝐶

Trigonometric Rules: Integrals of Trig Functions

\int c o s 𝑥 𝑑 𝑥 = s i n 𝑥 + 𝐶 \int s i n 𝑥 𝑑 𝑥 = - c o s 𝑥 + 𝐶 \int s e c 2 𝑥 𝑑 𝑥 = t a n 𝑥 + 𝐶 \int s e c 𝑥 t a n 𝑥 𝑑 𝑥 = s e c 𝑥 + 𝐶 \int c s c 2 𝑥 𝑑 𝑥 = - c o t 𝑥 + 𝐶

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 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 + 𝐶 \int 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 + 𝐶

You can of course practice typical problems below, each with a complete solution.

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. Finally add C."

We can express this process mathematically as

\int 𝑥 𝑛 𝑑 𝑥 = (1 𝑛 + 1) 𝑥 𝑛 + 1 + 𝐶 (𝑛 \neq - 1)

For example,

\int 𝑥 3 𝑑 𝑥 = (1 3 + 1) 𝑥 3 + 1 + 𝐶 (𝑛 \neq - 1) = 14 𝑥 4 + 𝐶

Power Rule Practice Problem #1

(a)

Find

\int 7 𝑑 𝑥 .

(b)

Find:

\int 𝑥 𝑑 𝑥 .

(c)

Find :

\int (7 + 𝑥) 𝑑 𝑥 .

(a) $7 𝑥 + 𝐶$
(b) $12 𝑥 2 + 𝐶$
(c) $7 𝑥 + 12 𝑥 2 + 𝐶$

[close solution]

Power Rule Practice Problem #2

(a)

Find:

\int 𝑥 2 𝑑 𝑥 .

(b)

Find:

\int 5 𝑥 4 𝑑 𝑥 .

(c)

Find:

\int (𝑥 2 - 5 𝑥 4) 𝑑 𝑥 .

(a) $13 𝑥 3 + 𝐶$
(b) $𝑥 5 + 𝐶$
(c) $13 𝑥 3 - 𝑥 5 + 𝐶$

[close solution]

Power Rule Practice Problem #3

Evaluate the integral:

\int (23 𝑤 5 - 14 𝑤 3 + 𝑤) 𝑑 𝑤 .

Power Rule Practice Problem #4

Evaluate the integral:

\int (𝑥 2 - 1) 2 𝑑 𝑥 .

Power Rule Practice Problem #5

Evaluate the integral:

\int 4 𝑥 - 7 𝑥 5 𝑥 3 𝑑 𝑥 .

Power Rule Practice Problem #6

(a)

Evaluate the integral:

\int \sqrt 𝑥 𝑑 𝑥 .

(b)

Evaluate the integral:

\int 1 \sqrt 𝑥 𝑑 𝑥 .

(c)

Evaluate the integral:

\int (\sqrt 𝑥 - 1 \sqrt 𝑥) 𝑑 𝑥 .

Power Rule Practice Problem #7

Find:

\int 𝑥 - 5 \sqrt 𝑥 𝑑 𝑥 .

Power Rule Practice Problem #8

Find

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

II. Exponential Rule: $𝑒 𝑥$

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

\int 𝑒 𝑥 𝑑 𝑥 = 𝑒 𝑥 + 𝐶

Exponential Rule

𝑒 𝑥

Practice Problem #1

Evaluate the integral:

\int 5 𝑒 𝑥 𝑑 𝑥 .

Exponential Rule

𝑒 𝑥

Practice Problem #2

Evaluate the integral:

\int (1 \sqrt 𝑥 5 - 𝑒 𝑥) 𝑑 𝑥 .

III. Exponential Rule: $𝑘 𝑥$ (where $𝑘$ is a constant)

This integral is also easy to remember: since $(𝑘 𝑥)' = 𝑘 𝑥 l n 𝑘$ , the integral of $𝑘 𝑥$ is also $𝑘 𝑥 l n 𝑘$

\int 𝑘 𝑥 𝑑 𝑥 = 1 l n 𝑘 𝑘 𝑥 + 𝐶

Exponential Rule

𝑘 𝑥

Practice Problem #1

Find:

\int 3 𝑥 𝑑 𝑥 .

Exponential Rule

𝑘 𝑥

Practice Problem #2

Find:

\int (𝑥 5 + 5 𝑥) 𝑑 𝑥 .

IV. Trigonometric Rules: Integrals of Trig Functions

We're listing here on the trig integrals that you should know at this early stage because each follows directly from a derivative you know. For example, since

(s i n 𝑥)' = c o s 𝑥

we know immediately that

\int c o s 𝑥 𝑑 𝑥 = s i n 𝑥 + 𝐶

Accordingly:

Trig Function Practice Problem #1

Find:

\int 4 c o s 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #2

Find:

\int 12 s i n 𝜃 𝑑 𝜃 .

Trig Function Practice Problem #3

Find:

\int 𝑑 𝑥 s e c 𝑥 .

Trig Function Practice Problem #4

Find:

\int s e c 2 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #5

Find:

\int s e c 𝑥 t a n 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #6

Find:

\int (c s c 2 𝑥 + 2 𝑥 2) 𝑑 𝑥 .

Trig Function Practice Problem #7

Find:

\int 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

Find:

\int t a n 2 𝑥 𝑑 𝑥 .

Trig Function Practice Problem #9

Find:

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

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Indefinite Integration

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

Exponential Rule: Integration of 𝑒𝑥

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