Integrating Trigonometric, Exponential & Reciprocal Functions (DP IB Analysis & Approaches (AA)): Revision Note

Written by: Paul

Reviewed by: Dan Finlay

Updated on 13 June 2025

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Integrating Trig Functions

How do I integrate sin, cos and sec^2?

The antiderivatives for sine and cosine are

$\int \sin x d x = - \cos x + c$

$\int \cos x d x = \sin x + c$

where $c$ is the constant of integration

Also, from the derivative of $\tan x$

$\int \sec^{2} x d x = \tan x + c$

Examiner Tips and Tricks

The standard integrals of $\sin x$ and $\cos x$ are both in the exam formula booklet.

The integral for $\sec^{2} x$ is not in the formula booklet. However the derivative result $f (x) = \tan x \Rightarrow f^{'} (x) = \sec^{2} x$ is in the formula booklet, so that can be used 'the other way round' to deduce the antiderivative.

For the linear function $a x + b$ , where $a$ and $b$ are constants,

$\int \sin (a x + b) d x = - \frac{1}{a} \cos (a x + b) + c$

$\int \cos (a x + b) d x = \frac{1}{a} \sin (a x + b) + c$

$\int \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + c$

For calculus with trigonometric functions angles must be measured in radians
- Ensure you know how to change the angle mode on your GDC

Examiner Tips and Tricks

Make sure you are familiar with the calculus formulas that are in the exam formula booklet. Many standard integrals are included, but remember that you can also use the standard derivatives in the booklet to help you find antiderivatives.

Just remember to add '+c', the constant of integration, for any indefinite integrals you solve.

Worked Example

a) Find, in the form $F (x) + c$ , an expression for each integral

$\int \cos x d x$
$\int \sec^{2} (3 x - \frac{π}{3}) d x$

b) A curve has equation $y = \int 2 \sin (2 x + \frac{π}{6}) d x$ .

The curve passes through the point with coordinates $(\frac{π}{3}, \sqrt{3})$ .

Find an expression for $y$ .

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Integrating e^x & 1/x

How do I integrate exponentials and 1/x?

The antiderivatives involving $e^{x}$ and $\ln x$ are

$\int e^{x} d x = e^{x} + c$

$\int \frac{1}{x} d x = \ln | x | + c$

where $c$ is the constant of integration

Examiner Tips and Tricks

Both of the standard integrals above are in the exam formula booklet.

For the linear function $(a x + b)$ , where $a$ and $b$ are constants,

$\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + c$

$\int \frac{1}{a x + b} d x = \frac{1}{a} \ln | a x + b | + c$

It follows from the last result (by using the using the reverse chain rule) that

$\int \frac{a}{a x + b} d x = \ln | a x + b | + c$

Examiner Tips and Tricks

With ln, it can sometimes be useful to write the constant of integration, $c$ , as a logarithm. I.e., by letting $c = \ln k$ for some (positive) constant $k$ .

Using the laws of logarithms, the answer can then be written as a single term. For example:

$\begin{array}{rcl} \int \frac{1}{x} d x & = & \ln | x | + c \\ = & \ln | x | + \ln k \\ = & \ln (k | x |) \end{array}$

Worked Example

A curve has the gradient function $f^{'} (x) = \frac{3}{3 x + 2} + e^{4 - x}$ .

Given the exact value of $f (1)$ is $\ln 10 - e^{3}$ find an expression for $f (x)$ .

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Integrating Trigonometric, Exponential & Reciprocal Functions (DP IB Analysis & Approaches (AA)): Revision Note

Integrating Trig Functions

How do I integrate sin, cos and sec^2?

Integrating e^x & 1/x

How do I integrate exponentials and 1/x?

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

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

1. Number & Algebra

Partial Fractions, Indices & Standard Form

Standard Form

Laws of Indices

Partial Fractions

Exponentials & Logs

Introduction to Logarithms

Laws of Logarithms

Solving Exponential Equations

Sequences & Series

Language of Sequences & Series

Sigma Notation

Arithmetic Sequences & Series

Geometric Sequences & Series

Applications of Sequences & Series

Compound Interest & Depreciation

Simple Proof & Reasoning

Proof by Deduction

Disproof by Counter Example

Proof by Induction & Contradiction

Proof by Induction

Proof by Contradiction

Binomial Theorem

Binomial Theorem

Binomial Coefficients & Pascal's Triangle

Extension of The Binomial Theorem

Permutations & Combinations

Counting Principles

Permutations

Combinations

Complex Numbers

Introduction to Complex Numbers

Operations with Complex Numbers

Modulus & Argument

Introduction to Argand Diagrams

Further Complex Numbers

Geometry of Complex Numbers

Modulus-Argument (Polar) Form

Exponential (Euler's) Form

Conversion between Forms of Complex Numbers

Complex Roots of Polynomials

De Moivre's Theorem

Proof of De Moivre's Theorem

Roots of Complex Numbers

Systems of Linear Equations

Systems of Linear Equations

Solving Systems Using Row Reduction

Number of Solutions to a System

2. Functions

Linear Functions & Graphs

Equations of a Straight Line

Parallel & Perpendicular Lines

Quadratic Functions & Graphs

Quadratic Functions

Factorising Quadratics

Completing the Square

Solving Quadratic Equations

Quadratic Inequalities

Discriminants

Functions Toolkit

Language of Functions

Composite Functions

Inverse Functions

Odd & Even Functions

Periodic Functions

Self-Inverse Functions

Graphing Functions & Their Key Features

Intersecting Graphs

Other Functions & Graphs

Exponential & Logarithmic Functions

Solving Equations Analytically

Solving Equations Graphically