Integration Rules

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Integration Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area underneath the graph of a function like this:

The integral of many functions are well known, and there are useful rules to work out the integral of more complicated functions, many of which are shown here.

There are examples below to help you.

Common Functions	Function	Integral
Constant	∫a dx	ax + C
Variable	∫x dx	x2/2 + C
Square	∫x2 dx	x3/3 + C
Reciprocal	∫(1/x) dx	ln|x| + C
Exponential	∫ex dx	ex + C
	∫ax dx	ax/ln(a) + C
	∫ln(x) dx	x ln(x) − x + C
Trigonometry (x in radians)	∫cos(x) dx	sin(x) + C
	∫sin(x) dx	-cos(x) + C
	∫sec2(x) dx	tan(x) + C
Rules	Function	Integral
Multiplication by constant	∫cf(x) dx	c∫f(x) dx
Power Rule (n≠-1)	∫xn dx	xn+1/(n+1) + C
Sum Rule	∫(f + g) dx	∫f dx + ∫g dx
Difference Rule	∫(f - g) dx	∫f dx - ∫g dx
Integration by Parts	See Integration by Parts
Substitution Rule	See Integration by Substitution

Examples

Example: what is the integral of sin(x) ?

From the table above it is listed as being −cos(x) + C

It is written as:

∫sin(x) dx = −cos(x) + C

Power Rule

Example: What is ∫x³ dx ?

The question is asking "what is the integral of x³ ?"

We can use the Power Rule, where n=3:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

∫x³ dx = x⁴/4 + C

Example: What is ∫√x dx ?

√x is also x^0.5

We can use the Power Rule, where n=½:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

∫x^0.5 dx = x^1.5/1.5 + C

Multiplication by constant

Example: What is ∫6x² dx ?

We can move the 6 outside the integral:

∫6x² dx = 6∫x² dx

And now use the Power Rule on x²:

= 6 x³/3 + C

Simplify:

= 2x³ + C

Sum Rule

Example: What is ∫cos x + x dx ?

Use the Sum Rule:

∫cos x + x dx = ∫cos x dx + ∫x dx

Work out the integral of each (using table above):

= sin x + x²/2 + C

Difference Rule

Example: What is ∫e^w − 3 dw ?

Use the Difference Rule:

∫e^w − 3 dw =∫e^w dw − ∫3 dw

Then work out the integral of each (using table above):

= e^w − 3w + C

Sum, Difference, Constant Multiplication And Power Rules

Example: What is ∫8z + 4z³ − 6z² dz ?

Use the Sum and Difference Rule:

∫8z + 4z³ − 6z² dz =∫8z dz + ∫4z³ dz − ∫6z² dz

Constant Multiplication:

= 8∫z dz + 4∫z³ dz − 6∫z² dz

Power Rule:

= 8z²/2 + 4z⁴/4 − 6z³/3 + C

Simplify:

= 4z² + z⁴ − 2z³ + C

Integration by Parts

See Integration by Parts

Substitution Rule

See Integration by Substitution