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

**Power Rule**

### Example: What is ∫x^{3} dx ?

The question is asking "what is the integral of x^{3 }?"

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

**Multiplication by constant**

### Example: What is ∫6x^{2} 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(x^{2}) 2x dx

dx^{2}/dx = 2x which cancels out the 2x here.

So ∫cos(x^{2}) 2x dx = sin(x^{2}) + C

Drafted by Eunice (Maths)

References:

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

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