Integration

Definition

Integration is used to find areas, volumes, central points and often is used to find the area underneath the graph of a function.

Common Functions of Integration

 FunctionIntegral
Constant∫ a dxax + c
Variable∫ x dx Common Functions of Integration Variable
Square∫ x2 dx  Common Functions of Integration Square
Reciprocal∫ (1/x) dx ln|x| + c
Exponential∫ ex dx ex + c
 ∫ ax dx Common Functions of Integration Exponential
 ∫ ln(x) dxx ln(x) − x + c
Trigonometry (x in radians)∫ cos(x) dx sin(x) + c
 ∫ sin(x) dx-cos(x) + c
 ∫ sec2(x) dxtan(x) + c

Rules of Integration

RulesFunctionIntegral
Multiplication by constant∫ cf(x) dxc∫f(x) dx
Power Rule (n≠-1)∫ xn dxRules of Integration Power Rule
Sum Rule∫ (f + g) dx∫f dx + ∫g dx
Difference Rule∫ (f – g) dx∫f dx – ∫g dx

Examples

RulesFunctionApplicationIntegral
Multiplication by constant∫ 5 cos(x) dx5 ∫ cos(x) dx5 sin(x) + c
Power Rule (n ≠ -1)∫ x2 dxRules of Integration Power Rule Example 1Rules of Integration Power Rule Example 2
Sum Rule∫ (cos(x) + x2) dx∫ cos(x) dx + ∫ x2 dxRules of Integration Sum Rule Example 1
Difference Rule∫ (x3 – sin(x)) dx∫ x3 dx – ∫ sin(x) dxRules of Integration Difference Rule Example 1

Learn More

Integration by Substitution (Reverse Chain Rule)

Common Derivatives

Differentiable

Implicit Differentiation

Calculus Index

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