Integrating Further Functions (DP IB Analysis & Approaches (AA)): Revision Note

Author

Paul

Last updated

6 December 2024

As with other problems in integration the results in this revision note may have further uses such as

evaluating a definite integral
finding the constant of integration
finding areas under a curve, between a line and a curve or between two curves

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Integrating with Reciprocal Trigonometric Functions

cosec (cosecant, csc), sec (secant) and cot (cotangent) are the reciprocal functions of sine, cosine and tangent respectively.

What are the antiderivatives involving reciprocal trigonometric functions?

$\int_{}^{} \sec^{2} x d x = \tan x + c$
$\int_{}^{} \sec x \tan x d x = \sec x + c$
$\int_{}^{} cosec x \cot x d x = - cosec x + c$
$\int_{}^{} {cosec}^{2} x d x = - \cot x + c$
These are not given in the formula booklet directly
- they are listed the other way round as ‘standard derivatives’
- be careful with the negatives in the last two results
- and remember “+c” !

How do I integrate these if a linear function of x is involved?

All integration rules could apply alongside the results above
The use of reverse chain rule is particularly common
- For linear functions the following results can be useful
  - $\int_{}^{} \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + c$
  - $\int_{}^{} \sec (a x + b) \tan (a x + b) d x = \frac{1}{a} \sec (a x + b) + c$
  - $\int_{}^{} cosec (a x + b) \cot (a x + b) d x = - \frac{1}{a} cosec (a x + b) + c$
  - $\int_{}^{} {cosec}^{2} (a x + b) d x = - \frac{1}{a} \cot (a x + b) + c$
- These are not in the formula booklet
  - they can be deduced by spotting reverse chain rule
  - they are not essential to remember but can make problems easier

Examiner Tips and Tricks

Even if you think you have remembered these antiderivatives, always use the formula booklet to double check
- those squares, negatives and "1 over"'s are easy to get muddled up!
Remember to use 'adjust' and 'compensate' for reverse chain rule when coefficients are involved

Worked Example

The graph of $y = f (x)$ where $f (x) = \int_{}^{} 2 \sec^{2} 5 x d x$ passes through the point $(\frac{π}{3}, 0)$ .

Show that $5 y = 2 (\sqrt{3} + \tan 5 x)$ .

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Integrating with Inverse Trigonometric Functions

arcsin, arccos and arctan are (one-to-one) functions defined as the inverse functions of sine, cosine and tangent respectively.

What are the antiderivatives involving the inverse trigonometric functions?

$\int_{}^{} \frac{1}{\sqrt{1 - x^{2}}} d x = \arcsin x + c$
$\int_{}^{} \frac{1}{1 + x^{2}} d x = \arctan x + c$
Note that the antiderivative involving $\arccos x$ would arise from

$\int_{}^{} - \frac{1}{\sqrt{1 - x^{2}}} d x = \arccos x + c$
- However, the negative can be treated as a coefficient of -1 and so
  $\int_{}^{} - \frac{1}{\sqrt{1 - x^{2}}} d x = - \int_{}^{} \frac{1}{\sqrt{1 - x^{2}}} d x = - \arcsin x + c$
- Similarly,

$\int_{}^{} \frac{1}{\sqrt{1 - x^{2}}} d x = - \int_{}^{} - \frac{1}{\sqrt{1 - x^{2}}} d x = - \arccos x + c$

Unless a question requires otherwise, stick to the first two results
These are listed in the formula booklet the other way round as ‘standard derivatives’
For the antiderivative involving $\arctan x$ , note that $(1 + x^{2})$ is the same as $(x^{2} + 1)$

How do I integrate these expressions if the denominator is not in the correct form?

Some problems involve integrands that look very similar to the above
- but the denominators start with a number other than one
- there are three particular cases to consider
The first two cases involve denominators of the form $a^{2} \pm {(b x)}^{2}$ (with or without the square root!)
- In the case $b = 1$ (i.e. denominator of the form $a^{2} \pm x^{2}$ ) there are two standard results
  - $\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} \arctan (\frac{x}{a}) + c$
  - $\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = \arcsin (\frac{x}{a}) + c, | x | < a$
  - Both of these are given in the formula booklet
  - Note in the first result, $a^{2} + x^{2}$ could be written $x^{2} + a^{2}$
- In cases where $b \neq 1$ then the integrand can be rewritten by taking a factor of $a^{2}$
  - the factor will be a constant that can be taken outside the integral
  - the remaining denominator will then start with 1
  - e.g. $9 + 4 x^{2} = 9 (1 + \frac{4}{9} x^{2}) = 9 (1 + {(\frac{2}{3} x)}^{2})$
The third type of problem occurs when the denominator has a (three term) quadratic
- i.e. denominators of the form $a x^{2} + b x + c$
  (a rearrangement of this is more likely)
  - the integrand can be rewritten by completing the square
  - e.g. $5 - x^{2} + 4 x = 5 - (x^{2} + 4 x) = 5 - [{(x + 2)}^{2} - 4] = 9 - {(x + 2)}^{2}$
    This can then be dealt with like the second type of problem above with " $x$ " replaced by " $x + 2$ "
  - This works since the derivative of $x + 2$ is the same as the derivative of $x$
    There is essentially no reverse chain rule to consider

Examiner Tips and Tricks

Always start integrals involving the inverse trig functions by rewriting the denominator into a recognisable form
- The numerator and/or any constant factors can be dealt with afterwards, using 'adjust' and 'compensate' if necessary

Worked Example

a) Find $\int_{}^{} \frac{1}{9 + x^{2}} d x$ .

b) Find $\int_{}^{} \frac{1}{\sqrt{5 - x^{2} + 4 x}} d x$ .

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Integrating Exponential & Logarithmic Functions

Exponential functions have the general form $y = a^{x}$ . Special case: $y = e^{x}$ .

Logarithmic functions have the general form $y = \log_{a} x$ . Special case: $y = \log_{e} x = \ln x$ .

What are the antiderivatives of exponential and logarithmic functions?

Those involving the special cases have been met before
- $\int_{}^{} e^{x} d x = e^{x} + c$
- $\int_{}^{} \frac{1}{x} d x = \ln | x | + c$
- These are given in the formula booklet
Also
- $\int_{}^{} a^{x} d x = \frac{1}{\ln a} a^{x} + c$
- This is also given in the formula booklet
By reverse chain rule
- $\int_{}^{} \frac{1}{x \ln a} d x = \log_{a} | x | + c$
- This is not in the formula booklet
  - but the derivative of $\log_{a} x$ is given
There is also the reverse chain rule to look out for
- this occurs when the numerator is (almost) the derivative of the denominator
- $\int_{}^{} \frac{f' (x)}{f (x)} d x = \ln |f (x)| + c$

How do I integrate exponentials and logarithms with a linear function of x involved?

For the special cases involving $e$ and $\ln$
- $\int_{}^{} e^{a x + b} d x = \frac{1}{a} e^{a x + b} + c$
- $\int_{}^{} \frac{1}{a x + b} d x = \frac{1}{a} \ln |a x + b| + c$
For the general cases
- $\int a^{p x + q} d x = \frac{1}{p \ln a} a^{p x + q} + c$
- $\int \frac{1}{(p x + q) \ln a} d x = \frac{1}{p} \log_{a} | p x + q | + c$
These four results are not in the formula booklet but all can be derived using ‘adjust and compensate’ from reverse chain rule

Examiner Tips and Tricks

Remember to always use the modulus signs for logarithmic terms in the antiderivative
- Once it is deduced that $g (x)$ in $\ln | g (x) |$ , say, is guaranteed to be positive, the modulus signs can be replaced with brackets

Worked Example

a) Show that $\int_{1}^{2} 4^{x} d x = \frac{6}{\ln 2}$ .

b) Find $\int \frac{1}{(2 x - 1) \ln 3} d x$ .

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Integrating Further Functions (DP IB Analysis & Approaches (AA)): Revision Note

Integrating with Reciprocal Trigonometric Functions

What are the antiderivatives involving reciprocal trigonometric functions?

How do I integrate these if a linear function of x is involved?

Integrating with Inverse Trigonometric Functions

What are the antiderivatives involving the inverse trigonometric functions?

How do I integrate these expressions if the denominator is not in the correct form?

Integrating Exponential & Logarithmic Functions

What are the antiderivatives of exponential and logarithmic functions?

How do I integrate exponentials and logarithms with a linear function of x involved?

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

Sign up now. It’s free!

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

1. Number & Algebra

Number & Algebra Toolkit

Standard Form

Laws of Indices

Partial Fractions

Exponentials & Logs

Introduction to Logarithms

Laws of Logarithms

Solving Exponential Equations

Sequences & Series

Language of Sequences & Series

Arithmetic Sequences & Series

Geometric Sequences & Series

Applications of Sequences & Series

Compound Interest & Depreciation

Simple Proof & Reasoning

Proof

Proof by Induction & Contradiction

Proof by Induction

Proof by Contradiction

Binomial Theorem

Binomial Theorem

Extension of The Binomial Theorem

Permutations & Combinations

Counting Principles

Permutations & Combinations

Complex Numbers

Introduction to Complex Numbers

Modulus & Argument

Introduction to Argand Diagrams

Further Complex Numbers

Geometry of Complex Numbers

Forms of Complex Numbers

Complex Roots of Polynomials

De Moivre's Theorem

Roots of Complex Numbers

Systems of Linear Equations

Systems of Linear Equations

Algebraic Solutions

2. Functions

Linear Functions & Graphs

Equations of a Straight Line

Quadratic Functions & Graphs

Quadratic Functions

Factorising & Completing the Square

Solving Quadratics

Quadratic Inequalities

Discriminants

Functions Toolkit

Language of Functions

Composite & Inverse Functions

Symmetry of Functions

Graphing Functions

Other Functions & Graphs

Exponential & Logarithmic Functions

Solving Equations

Modelling with Functions

Reciprocal & Rational Functions

Reciprocal & Rational Functions

Transformations of Graphs

Translations of Graphs

Reflections of Graphs

Stretches of Graphs

Composite Transformations of Graphs

Polynomial Functions

Factor & Remainder Theorem

Polynomial Division