## Integration in C3 and C4

__Integration by Inspection/Reverse Chain Rule__

You cannot integrate a Trigonometric function with a power greater than one. You cannot add one to the power and divide by the power, you need to get rid of the power using an identity. Three important identities for integrating Trigonometric functions are:

__Integration by Substitution__- Differentiate the substitute and rearrange to make dx the subject.
- Substitute x and dx at the same time and simplify the expression.
- If there are any more x terms left then rearrange the substitute to get that x term and substitute it into the integral, so you have an integral with only u and du.
- Integrate with respect to u then substitute x back into you answer or change the limits from x to the limits for u using the substitute equation. Do whichever one is easiest.

__Integrating Trigonometric Functions__You cannot integrate a Trigonometric function with a power greater than one. You cannot add one to the power and divide by the power, you need to get rid of the power using an identity. Three important identities for integrating Trigonometric functions are:

__Compound angle identities from the formula booklet__

__Example__

__Integration by Parts__- Always label the term that will simplify when you differentiate it as u, for example 4x to 4 when you differentiate it.
- If you are using by parts to integrate lnx, then you have to label it as u because you cannot integrate lnx to get v if you label it as
- If you do not have two terms being multiplied to use Integration by parts, you can always use 1 as a term, because 1 multiplied by anything is itself.
- You might need to use integration by parts twice for some questions.

__Integrating Fractions__- Always convert the fraction into index form and then integrate it normally if it has a power that is not -1.

- If it has a power of -1 then you cannot integrate it normally because you cannot add 1 and then divide by zero. You have to use the following rule:

- To get the derivative of the denominator in the numerator you might need to take out a factor.

- If you cannot get the derivative of the denominator in the numerator by taking out a factor, then you need to use partial fractions to simplify the denominator of the fraction so that you can then get the derivative of the denominator in the numerator, to then use the rule above.

__Integrating five different types of fractions:__

Using the reverse chain rule:

Using the natural log method:

Splitting the numerator:

Using Partial Fractions:

Using Integration by substitution:

__Volume of Revolution__

__Integrating Parametric equations__