Calculus Integrals Math Sheet

An Engineers Quick Calculus Integrals Reference

Integrals

Definition of an Integral

The integral is a mathematical analysis applied to a function that results in the area bounded by the graph of the function, x axis, and limits of the integral. Integrals can be referred to as anti-derivatives, because the derivative of the integral of a function is equal to the function.

Properties

the integral of the addition or subtraction of two functions

the integral of a function where the two points of evaluation are the same

the integral of a function where the two points of evaluation are swapped

the integral of a function multiplied by a constant

Common Integrals

integral of a constant

integral of a variable raised to a power

integral of a x raised to -1 or 1 over x

integral of linear function raised to -1

integral of natural log function

integral of e to the x

integral of cosine function

integral of sine function

integral of secant squared function

integral of secant tangent function

integral of cosecant cotangent function

integral of cosecant squared function

integral of tangent function

integral of secant function

common integral with inverse tangent

common integral with inverse sine

Integration by Substitution

integration by u substitution

integral by u substitution

Integration by Parts

integration by parts

integral by parts

Integration by Trigonometric Substitution

Trigonometric identities can be use with integration substitution to simplify integrals. There are three common substitutions.

First Trigonometric Substitution

first trignometric substitution

To take advantage of the property

integral by trig substitution

Substitute

substitute sin function

substitute cosine function for dx

After substitution

result of first trig substitution

Second Trigonometric Substitution

second trignometric substitution

To take advantage of the property

integral by trig substitution- secant squared

Substitute

integration substition for x - secant

trig substitution for dx

After substitute

result for second trig substitution

Third Trigonometric Substitution

third trigonometric substitution

To take advantage of the property

integral by trig substitution - tangent squared

Substitute

integration substitution for x - tangent

trig substitution for dx - secant squared

After substitute

result for third trig substitution

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