Let's learn about the rules to intergration in A-Level Maths!

Power Rule

Example: What is ∫x³ dx ? 

The question is asking "what is the integral of x³ ?" 

We can use the Power Rule, where n=3:

Multiplication by constant

Example: What is ∫6x² dx ? 

We can move the 6 outside the integral:

Trigonometry (x in radians)

Example: what is the integral of sin(x) ? 

From the table above it is listed as being −cos(x) + C 

It is written as:

Sum Rule

Example:

What is ∫(cos x +  x)  dx ? 

Use the Sum Rule:

Reciprocal rule

Integration by Parts

∫u v dx = u∫v dx −∫u' (∫v dx) dx  

• u is the function u(x) 
• v is the function v(x) 
• u' is the derivative of the function u(x) 

Exponential rule

Integration by Substitution

Note that we have g(x) and its derivative g'(x)

Example: 

∫cos(x²) 2x dx 

dx²/dx = 2x  which cancels out the 2x here.

So ∫cos(x²) 2x dx = sin(x²) + C

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Drafted by Eunice (Maths)

References:

https://www.mathsisfun.com/calculus/integration-by-parts.html

https://www.mathsisfun.com/calculus/integration-rules.html