Integration (FP3)

Q: Do I need to memorise all standard integration results for FP3?

No, the Edexcel formula book provides all standard hyperbolic, inverse trigonometric and surd integration results. You only need to recall integration by parts and substitution techniques, and how to derive reduction formulae.

Q: Can polar equations be used for arc length or surface area questions in FP3?

No, polar form arc length and surface area are explicitly out of scope for FP3. All questions will use Cartesian or parametric equations only.

Edexcel International A-Level Further Mathematics· FP3 4.1 to 4.6, 2018 spec Issue 3· 25 min read

1. Integrating Hyperbolic, Inverse Trigonometric and Hyperbolic Functions★★☆☆☆⏱ 5 min

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Standard hyperbolic function integral results are provided in the formula book, so you can apply them directly to evaluate indefinite and definite integrals. For inverse trigonometric and inverse hyperbolic functions, use integration by parts with $u$ set equal to the inverse function, and $d v = d x$ .

📘 Definition

Integration of inverse functions

For any integrable inverse function $f^{- 1} (x)$ , use the integration by parts setup: $u = f^{- 1} (x)$ , $d v = d x$ , so $d u = \frac{1}{f ^{'} ( f ^{- 1} ( x ))} d x$ , $v = x$ , giving $\int f^{- 1} (x) d x = x f^{- 1} (x) - \int \frac{x}{f ^{'} ( f ^{- 1} ( x ))} d x$ .

📐 Worked Example

Evaluate $\int_{0}^{1} arsinh x d x$

1
Set $u = arsinh x$ , $d v = d x$
2
$du = \frac{1}{\sqrt{1+x^2}}dx, v = x$
3
Apply integration by parts formula:
4
$\int \text{arsinh }x dx = x \text{arsinh }x - \int \frac{x}{\sqrt{1+x^2}}dx$
5
Evaluate the remaining integral using substitution $t = 1 + x^{2}$ , $d t = 2 x d x$ :
6
$\int \frac{x}{\sqrt{1+x^2}}dx = \sqrt{1+x^2} + C$
7
Apply limits 0 to 1:
8
$[x \text{arsinh }x - \sqrt{1+x^2}]_0^1 = \text{arsinh }1 - \sqrt{2} + 1$
9
Use standard result $arsinh 1 = ln (1 + 2)$ to get the final value: $ln (1 + 2) - 2 + 1 \approx 0.467$

2. Trigonometric and Hyperbolic Substitution for Integrals★★★☆☆⏱ 5 min

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Use standard substitutions to simplify integrals involving quadratic surds. All standard substitutions and corresponding integral results are given in the formula book. For more complex surd integrals, the required substitution will be provided in the question, so you do not need to invent new substitutions.

Integrand Form	Substitution	Simplified Surd
$1/ a^{2} + x^{2}$	$x = a sinh u$	$a cosh u$
$1/ a^{2} - x^{2}$	$x = a sin θ$	$a cos θ$
$1/ x^{2} - a^{2}$	$x = a cosh u$	$a sinh u$
$1/ (a^{2} + x^{2})$	$x = a tan θ$	$a^{2} sec^{2} θ$

📐 Worked Example

Evaluate $\int \frac{1}{4 x ^{2} - 9} d x$ for $x > 3/2$

1
Rewrite the integrand to match the standard form:
2
$\int \frac{1}{\sqrt{(2x)^2 - 3^2}}dx$
3
Use substitution $2 x = 3 cosh u$ , so $2 d x = 3 sinh u d u$ , $d x = \frac{3}{2} sinh u d u$
4
Substitute into the integral:
5
$\int \frac{1}{\sqrt{9 \cosh^2 u - 9}} \times \frac{3}{2}\sinh u du = \int \frac{1}{3 \sinh u} \times \frac{3}{2}\sinh u du = \int \frac{1}{2} du$
6
$= \frac{1}{2}u + C = \frac{1}{2}\text{arcosh}\left(\frac{2x}{3}\right) + C$
7
This matches the standard result from the formula book, so you could also write it directly as $\frac{1}{2} ln (2 x + 4 x^{2} - 9) + C$

3. Deriving and Applying Reduction Formulae★★★★☆⏱ 6 min

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📘 Definition

Reduction formula

A recurrence relation that expresses $I_{n}$ (the integral of a function raised to power $n$ ) in terms of $I_{n - k}$ for $k < n$ , usually $k = 2$ for trigonometric and hyperbolic functions.

📐 Worked Example

Derive a reduction formula for $I_{n} = \int_{0}^{π /2} sin^{n} x d x$ , and use it to evaluate $I_{4}$

1
Split the integrand into two terms for integration by parts:
2
$I_n = \int_0^{\pi/2} \sin^{n-1}x \times \sin x dx$
3
Set $u = sin^{n - 1} x$ , $d v = sin x d x$ , so $d u = (n - 1) sin^{n - 2} x cos x d x$ , $v = - cos x$
4
Apply integration by parts:
5
$I_n = [-\sin^{n-1}x \cos x]_0^{\pi/2} + (n-1)\int_0^{\pi/2} \sin^{n-2}x \cos^2 x dx$
6
The boundary term evaluates to 0, substitute $cos^{2} x = 1 - sin^{2} x$ :
7
$I_n = (n-1)\int_0^{\pi/2} \sin^{n-2}x (1 - \sin^2 x) dx = (n-1)I_{n-2} - (n-1)I_n$
8
Rearrange to get the recurrence relation:
9
$I_n + (n-1)I_n = (n-1)I_{n-2} \implies nI_n = (n-1)I_{n-2}$
10
Evaluate $I_{4}$ using base case $I_{0} = \int_{0}^{π /2} 1 d x = π /2$ :
11
$I_4 = \frac{3}{4}I_2 = \frac{3}{4} \times \frac{1}{2}I_0 = \frac{3}{8} \times \frac{\pi}{2} = \frac{3\pi}{16}$

4. Calculating Arc Length for Cartesian and Parametric Curves★★★☆☆⏱ 4 min

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Arc length formulae for Cartesian and parametric curves are given in the formula book. Polar form arc length is out of scope for FP3. Always simplify the expression inside the square root first, as it often simplifies to a perfect square.

📐 Worked Example

Calculate the arc length of the curve $y = \frac{2}{3} x^{3/2}$ between $x = 0$ and $x = 3$

1
Differentiate y with respect to x:
2
$\frac{dy}{dx} = x^{1/2} = \sqrt{x}$
3
Substitute into the Cartesian arc length formula:
4
$s = \int_0^3 \sqrt{1 + (\frac{dy}{dx})^2} dx = \int_0^3 \sqrt{1 + x} dx$
5
Evaluate the integral:
6
$s = \left[\frac{2}{3}(1+x)^{3/2}\right]_0^3 = \frac{2}{3}(4^{3/2} - 1^{3/2}) = \frac{2}{3}(8 - 1) = \frac{14}{3} \approx 4.67$

5. Calculating Surface Area of Revolution★★★★☆⏱ 5 min

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Surface area of revolution about the x-axis is given by $S_{x} = 2 π \int y d s$ , where $d s$ is the arc length element matching the curve's form (Cartesian or parametric). Do not forget the $y$ term or the $d s$ term, as these are common sources of error.

📐 Worked Example

Find the surface area formed when the curve $y = x$ for $0 \leq x \leq 4$ is rotated $2 π$ radians about the x-axis

1
Differentiate y with respect to x:
2
$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
3
Write the ds term for Cartesian form:
4
$ds = \sqrt{1 + (\frac{dy}{dx})^2} dx = \sqrt{1 + \frac{1}{4x}} dx = \sqrt{\frac{4x + 1}{4x}} dx = \frac{\sqrt{4x + 1}}{2\sqrt{x}} dx$
5
Substitute into the surface area formula:
6
$S_x = 2\pi \int_0^4 y ds = 2\pi \int_0^4 \sqrt{x} \times \frac{\sqrt{4x + 1}}{2\sqrt{x}} dx$
7
Simplify the integrand, the $x$ terms cancel out:
8
$S_x = \pi \int_0^4 \sqrt{4x + 1} dx$
9
Evaluate the integral using substitution $u = 4 x + 1$ , $d u = 4 d x$ :
10
$S_x = \pi \times \frac{1}{4} \times \frac{2}{3}[(4x + 1)^{3/2}]_0^4 = \frac{\pi}{6}(17^{3/2} - 1) \approx 36.2$

6. Common Pitfalls

Wrong move:

Using $x = a sinh u$ substitution for $x^{2} - a^{2}$ instead of $x = a cosh u$

Why:

$sinh^{2} u + 1 = cosh^{2} u$ , so $a^{2} sinh^{2} u + a^{2} = a cosh u$ , which works for $x^{2} + a^{2}$ not $x^{2} - a^{2}$

Correct move:

Use $x = a cosh u$ for $x^{2} - a^{2}$ , as $cosh^{2} u - 1 = sinh^{2} u$ , simplifying the surd to $a sinh u$ for $x \geq a$

Wrong move:

Writing $S_{x} = 2 π \int y d x$ instead of $2 π \int y d s$ for surface area calculations

Why:

Surface area depends on the length of the curve segment, not just the x-axis interval, so omitting $d s$ gives an incorrect value

Correct move:

Always substitute the correct $d s$ form (Cartesian or parametric) matching the curve's equation before evaluating the integral

Wrong move:

Splitting $I_{n} = \int sin^{n} x d x$ as $u = sin^{n} x$ , $d v = d x$ when deriving reduction formulae

Why:

This split leads to an integral with $x sin^{n - 1} x cos x$ , which is more complex and does not produce a lower power of $sin x$

Correct move:

Split powers of trigonometric/hyperbolic functions into a $(n - 1)$ power term and a 1 power term for integration by parts to get a recurrence relation with $I_{n - 2}$

Wrong move:

Trying to integrate inverse functions directly instead of using integration by parts

Why:

Inverse functions have no elementary antiderivative when integrated directly, so this approach will not work

Correct move:

For integrals of inverse functions like $\int arctan x d x$ , set $u =$ inverse function, $d v = d x$ , then evaluate the resulting simpler integral

Wrong move:

Using incorrect limits for parametric arc length/surface area integrals

Why:

The parameter limits must correspond exactly to the start and end points of the curve segment being measured, not the x-axis limits

Correct move:

Check that you are using the correct parameter interval that maps to the curve endpoints given in the question

7. Quick Reference Cheatsheet

Concept	Formula / Technique	Exam Tip
Integrate inverse functions	Use by parts: $u = f^{- 1} (x)$ , $d v = d x$	Use formula book surd results directly for the resulting integral to save time
$a^{2} + x^{2}$ substitution	$x = a sinh u$	Simplifies to $a cosh u$ , result maps to $arsinh (x / a)$
$a^{2} - x^{2}$ substitution	$x = a sin θ$	Simplifies to $a cos θ$ , result maps to $arcsin (x / a)$
$x^{2} - a^{2}$ substitution	$x = a cosh u$	Simplifies to $a sinh u$ , result maps to $arcosh (x / a)$
Reduction formula derivation	Use by parts: $u = f (x)^{n - 1}$ , $d v = f (x) d x$	State the recurrence relation explicitly before applying to base cases
Cartesian arc length	$s = \int 1 + (d y / d x)^{2} d x$	Simplify the term inside the square root first, look for perfect squares
Parametric arc length	$s = \int (d x / d t)^{2} + (d y / d t)^{2} d t$	Use parameter limits matching the curve endpoints, not x/y limits
Surface area (x-axis)	$S_{x} = 2 π \int y d s$	Use the $d s$ form matching the curve's equation (Cartesian/parametric)

8. Frequently Asked

Do I need to memorise all standard integration results for FP3?

No, the Edexcel formula book provides all standard hyperbolic, inverse trigonometric and surd integration results. You only need to recall integration by parts and substitution techniques, and how to derive reduction formulae.

Can polar equations be used for arc length or surface area questions in FP3?

No, polar form arc length and surface area are explicitly out of scope for FP3. All questions will use Cartesian or parametric equations only.

Going deeper

formula_bookEdexcel IAL Further Mathematics Formula BookRefer to FP3 Integration, Arc Length and Surface Area sections for given formulae

What's Next

Now that you have mastered FP3 integration techniques, you are ready to apply these skills to the remaining core FP3 topics, as well as synoptic questions across the Further Mathematics specification. Integration of hyperbolic functions and reduction formulae are often combined with differential equations topics in FP4, while arc length and surface area calculations may appear in synoptic questions linking to coordinate geometry in FP2. Be sure to practice full past paper questions for this topic, as exam questions typically combine multiple integration techniques in a single multi-part problem, worth 8-12 marks. You should also revise integration techniques from Pure 3 to ensure you can quickly recall substitution and by parts rules under timed exam conditions.