AP®MathsCollege BoardCalculus ABStudy GuidesUnit 6: Integration & Accumulation of ChangeIntegration & AntiderivativesIndefinite Integral Rules

Indefinite Integral Rules (College Board AP® Calculus AB): Study Guide

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 12 May 2026

Indefinite integrals of sums, differences and constant multiples

How do I integrate sums, differences and constant multiples of terms?

When integrating sums or differences of terms
- the integral is simply the sum (or difference) of the integrals of the terms
- This may be expressed as $\int (f (x) \pm g (x)) d x = \int f (x) d x \pm \int g (x) d x$
  - E.g. $\int (x^{2} + \cos x) d x = \int x^{2} d x + \int \cos x d x = \frac{1}{3} x^{3} + \sin x + C$
  - We still only need a single constant of integration
- Note that products and quotients of terms cannot be integrated in this way
  - They may need to be expanded or simplified first
  - Or a more advanced integration technique may need to be used
    - See the 'Methods of Integration' study guide
When integrating a constant multiple of a term
- the term may be brought out in front of the integral as a multiplier
- This may be expressed as $\int k f (x) d x = k \int f (x) d x$
  - E.g. $\int 5 e^{x} d x = 5 \int e^{x} d x = 5 e^{x} + C$
The rules for sums, differences and constant multiples can be combined
- If $p$ and $q$ are constants, then
  - $\int (p f (x) \pm q g (x)) d x = p \int f (x) d x \pm q \int g (x) d x$
  - This idea can be extended for any number of terms inside the integral

Worked Example

Find the indefinite integral $\int (3 x^{3} - 2 \sin 3 x + 5 e^{4 x}) d x$ .

Answer:

Write as a sum or difference of integrals, with constants pulled in front as multipliers

$3 \int x^{3} d x - 2 \int \sin 3 x d x + 5 \int e^{4 x} d x$

Now integrate the terms one by one

$3 \cdot \frac{1}{4} x^{4} - 2 \cdot (- \frac{1}{3} \cos 3 x) + 5 \cdot \frac{1}{4} e^{4 x} + C$

Multiply out and simplify

$\frac{3}{4} x^{4} + \frac{2}{3} \cos 3 x + \frac{5}{4} e^{4 x} + C$

Examiner Tips and Tricks

Once you have done a lot of practice, you should be able to skip the step of writing down the separate integrals such as $3 \int x^{3} d x - 2 \int \sin 3 x d x + 5 \int e^{4 x} d x$ .

Simplifying expressions to find indefinite integrals

How can I simplify expressions to help me find indefinite integrals?

Sometimes expressions need to be simplified before you can integrate them
This may involve expanding brackets
- E.g. $\int {(x^{2} + 2)}^{2} d x$
- Expand the brackets
  - ${(x^{2} + 2)}^{2} = x^{4} + 4 x^{2} + 4$
- Now integrate in the usual way
  - $\int {(x^{2} + 2)}^{2} d x = \int (x^{4} + 4 x^{2} + 4) d x = \frac{1}{5} x^{5} + \frac{4}{3} x^{3} + 4 x + C$
Or rearranging fractions (including using laws of exponents)
- E.g. $\int \frac{5 x^{3} - 3}{x^{2}} d x$
- Rearrange the fraction
  - $\frac{5 x^{3} - 3}{x^{2}} = \frac{5 x^{3}}{x^{2}} - \frac{3}{x^{2}} = 5 x - \frac{3}{x^{2}}$
- Use laws of exponents
  - $5 x - \frac{3}{x^{2}} = 5 x - 3 x^{- 2}$
- Now integrate in the usual way
  - $\int \frac{5 x^{3} - 3}{x^{2}} d x = \int (5 x - 3 x^{- 2}) d x = \frac{5}{2} x^{2} + 3 x^{- 1} + C$

Worked Example

Find the indefinite integral $\int \frac{2 x^{3} + 5}{\sqrt{x}} d x$ .

Answer:

Simplify the fraction

$\begin{array}{rcl} \int \frac{2 x^{3} + 5}{\sqrt{x}} d x & = & \int \frac{2 x^{3} + 5}{x^{\frac{1}{2}}} d x \\ = & \int (\frac{2 x^{3}}{x^{\frac{1}{2}}} + \frac{5}{x^{\frac{1}{2}}}) d x \\ = & \int (2 x^{\frac{5}{2}} + 5 x^{- \frac{1}{2}}) d x \end{array}$

Separate the integral

$2 \int x^{\frac{5}{2}} d x + 5 \int x^{- \frac{1}{2}} d x$

Use the antiderivatives

$2 \cdot \frac{1}{\frac{7}{2}} x^{\frac{7}{2}} + 5 \cdot \frac{1}{\frac{1}{2}} x^{\frac{1}{2}} + C$

Don't forget the constant of integration

Simplify

$\frac{4}{7} x^{\frac{7}{2}} + 10 x^{\frac{1}{2}} + C$

Examiner Tips and Tricks

Don't forget to include the constant of integration. It is a common mistake reported by readers.

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Indefinite Integral Rules (College Board AP® Calculus AB): Study Guide

Indefinite integrals of sums, differences and constant multiples

How do I integrate sums, differences and constant multiples of terms?

Worked Example

Examiner Tips and Tricks

Simplifying expressions to find indefinite integrals

How can I simplify expressions to help me find indefinite integrals?

Worked Example

Examiner Tips and Tricks

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Dan Finlay