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Calculus Integrals Math Sheet

An Engineers Quick Calculus Integrals Reference

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$calculus integrals sheet$

Integrals Definition of an Integral Properties Common Integrals Integration by Subs. Integration by Parts Integration by Trig. Subs. Second Trig. Subs. Third Trig. Subs.

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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