Integration

Overview

The Fundamental Theorem of Calculus states that integration is the reverse of differentiation.

An indefinite integral has no limits, and is an infinite family of functions differing by some constant of integration, . The constant of integration can be found if a value of for a given is known.

The definite integral is found by evaluating the integrated expression at the upper limit and subtracting the integrated expression evaluated at the lower limit . This gives the signed area. If the integrand changes sign between the two limits, then the interval needs to be split to find the total area.

The area between two curves can be found with a single integral. For two curves ) and , provided that for (the curves do not intersect), the area bounded by the two curves,, and is given by:

This can be shown visually:

As with the area between the function and the axis, this area is signed, so if the functions intersect, then multiple integrals must be considered.

Integration rules

Integration of polynomials

For all rational :

Reverse chain rule

Consider differentiating a power of a function:

We can reverse this to get a general rule for integration (given):

There is a special case for when (given):

Integration by parts

Integration by parts allows for integration of products of functions. The formula is:

Generally, should be chosen to get simpler when differentiated, and should be chosen to not get any more complicated when integrated. A general rule for choosing is ILATE:

Inverse trigonometric functions
Logarithmic functions
Algebraic functions
Trigonometric functions
Exponential functions

At A-level, repeated application of integration by parts may be required. Also, integration by parts can be used to find integrals of functions such as , by setting and :

Derivation

IS?: Learners should understand the relationship between [integration by parts] and the product rule.
Integration by parts comes from integrating an application of the product rule:

Integration by substitution

To integrate by substitution:

Define a new variable.
Rewrite the integrand in terms of the new variable.
Make use of the chain rule to change the integral from an integral in terms of the old variable into an integral in terms of the new variable.
If the integral has limits, change the limits on the integral from limits for the old variable to limits for the new variable.
Carry out the integration.
Substitute for the original variable if needed.

Integration using trig identities

Pythagorean identities are used to integrate and :

Double angle identities are used to integrate and :

Integration by partial fractions

For some rational functions, partial fractions can be used to put them in a form that allows for integration:

Integration as the limit of a sum

IS: learners should know that the area under a graph can be found as the limit of a sum of areas of rectangles.
Integration can be defined as the limit of a sum. This can be expressed using the following formula:

This process can also be shown graphically as the sum of areas of rectangles: