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Integration Techniques Made Easy
Introduction - Integration Techniques Made Easy
When you start A Level Maths in year 12. It probably won’t be until after January when you start Calculus. This covers differentiation and integration. Integration is what we are going to discuss here and it is the reverse of differentiation. As well as it being a part of pure maths it is also seen in mechanics. Here we are going to look at some of the techniques, and provide short examples.
Just what is Integration?
As mentioned above integration is the reverse of differentiation. You can find the equation of a curve given the gradient function. It is also used to calculate the area under a curve or which would be important in the word of engineering.
For example, if y = x^2, the integral is:
\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C.
Here, C is a constant, and this must be included when performing an indefinite integral.
There are many practical applications of integration such as:
- Calculating areas under curves.
- Working with motion problems in mechanics where acceleration is not constant.
Solving probability distributions in statistics.
The Basic Rules of Integration
Let us just recap a few basic rules that you should know:
1. Power rule:
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,
2. Constant multiple rule:
\int a\,f(x)\,dx \;=\; a \int f(x)\,dx.
3. Sum and difference rule:
\int a\,f(x)\,dx \;=\; a \int f(x)\,dx.
\int \bigl[f(x) – g(x)\bigr]\,dx \;=\; \int f(x)\,dx \;-\; \int g(x)\,dx.
4. Exponential functions:
\int e^x\,dx = e^x + C,
For rule 4 – remember that the integral of e^x does not follow the standard process where 1 is reduced from the original power.
These rules are the foundation of many A-Level maths integration questions.
Substitution Method
It is important when to know which particular method of integration is needed and when. Quite often the question will tell. But there are times you will need to determine the correct method.
The method of
Substitution is used when one function is nested inside another.
Steps:
- Let u = f(x), where f(x) is the inner function.
- Differentiate u to find [/latex]du[/latex].
- Replace the original integral in terms of u.
- Integrate with respect to u.
- Substitute back in terms of u.
Example:
2x(x^2 + 1)^5
- Let u = x^2 + 1. which after differentiating gives du = 2x\,dx.
- The integral then becomes \int 2x\,(x^2 + 1)^5 \,dx = \int u^5 \,du.
- Substitute your answer back for x: \boxed{\frac{(x^2 + 1)^{6}}{6} + C}.
Using Integration by Substitution really helps to simplify integrals that look complicated at first glance.
Integration by Parts
Make sure you know when to use Integration By Parts and when to use integration by substitution.
You will use integration by parts when you are dealing with a product of functions.
Integration by parts is for products of functions.
\boxed{\int u\,dv \;=\; u\,v \;-\;\int v\,du \;+\; C}
Steps:
Generally take the easier function to be u as it is easier to differentiate. This will make the dv easier to integrate.
Carefully differentiate u to give du
Integrate dv to give v
Apply the above to the formula
Example:
\int x e^x\,dx
- This is a product of functions so clearly you would use integration by parts.
- Pick x as this is the easier function to differentiate and dv = e^x\,dx
- Apply formula to get:
You will use Integration by parts in a number of areas such as calculus, mechanics, and exponential problems.
Partial Fractions
If you have a completed algebraic fraction then you can split this into Partial Fractions in order to perform an integral.
Steps:
- Always factorise the denominator and look out for repeated linear factors.
- Then split the single algebraic fraction into multiple fractions.
- You can then integrate each term separately.
Example:
\int \frac{3x + 5}{(x+1)(x+2)} \, dx
- First split the algebraic fraction: \frac{3x+5}{(x+1)(x+2)}= \frac{A}{x+1} + \frac{B}{x+2}.
- Find the values of A and B
- You can then integrate each term as follows:
\int \frac{2}{x+1}\,dx + \int \frac{1}{x+2}\,dx = 2\ln|x+1| + \ln|x+2| + C.
Partial fractions allow us to simplify integrals that would otherwise be very difficult.
Definite Integrals
When working with a definite integral you will obtain a specific numerical value. There should be no +C. You will use a definite integral when finding the area under a curve.
Definite integrals calculate the area between limits ( a ) and ( b ):
\int_{a}^{b} f(x)\,dx. Steps:
In order to find the indefinite integral
- Find the integral
- Substitute the upper and lower limits and subtract
\boxed{\int_{a}^{b} f(x)\,dx = F(b) – F(a)}.
Example:
Find \int_{0}^{2} x^{2}\,dx.
The integral is: F(x)=\frac{x^{3}}{3},
When performing the substitution of the upper and lower limits:
\int_{0}^{2} x^{2}\,dx \;=\; F(2)\;-\;F(0).
Then finally perform the correct arithmetic:
\boxed{\frac{8}{3}}You will use definite integrals in, mechanics and probability.
Tips for mastering integrals
Learn the basic rules – power, exponentials, logarithms.
Practice substitution and integration by parts. Factorise or give a variable name to get in easily.
Normal fractions can also be dealt with by partial fractions.
Always show your workings in examinations – you will be given credit for clearly stating calculations.
Check your answer by differentiating again to see if you get back the original.
Integration appears on all the major A-Level exam boards (e.g. AQA, Edexcel, OCR, MEI). Mastering these will enable you to score more marks in calculus, mechanics, statistics, problem-solving etc.
Ready to boost your exam confidence?
If you are looking for more indepth help regarding exams, exam technique and tackling harder exam questions explore our half term 3-Day A-Level Maths Revision Course which takes place online.
Author Bio – S. Mahandru
S. Mahandru • Head of Maths, Exam.tips
S. Mahandru has over 15 years’ experience teaching A-Level Maths in UK classrooms. He is a published author and an approved A-Level examiner. S. Mahandru specialises in calculus, integration, mechanics, and statistics, helping students understand concepts clearly and improve their grades.
🧭 Next topic:
“Next, refresh the core differentiation ideas that support integration.”
FAQS
What is the difference between differentiation and integration?
Differentiation allows you to find the rate of change or the gradient function of a curve.
Integration is the reverse process. It allows us to find the original equation or more commonly to find the area under a curve.
How can I tell which integration method to use in an exam question?
Carefully at the expression that you need to integrate:
If there is a function within another → it is Substitution
If it is the product of two functions → it is Integration by Parts
It it is a algebraic fraction → it is Partial Fractions
It is important to be practising different question types which will help you identify which technique to apply.
Why is the constant "C" important in indefinite integrals?
The constant represents a constant value that could have been present prior to differentiation.
Once you differentiate a function, the constant is removed. Adding the constant allows that the general solution can account for a particular solution.