Integration Rules - Math Is Fun

Integration Rules

Integration

Integration can be used to find areas, volumes, central points and many useful things. It is often used to find the area underneath the graph of a function and the x-axis.

The first rule to know is that integrals and derivatives are opposites!

Sometimes we can work out an integral, because we know a matching derivative.

Integration Rules

Here are the most useful rules, with examples below:

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+1n+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

Example: what is the integral of 1/x ?

From the table above it is listed as being ln|x| + C

It is written as:

∫(1/x) dx = ln|x| + C

The vertical bars || either side of x mean absolute value, because we don't want to give negative values to the natural logarithm function ln.

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+1n+1 + C

∫x3 dx = x44 + C

Example: What is ∫√x dx ?

√x is also x0.5

We can use the Power Rule, where n=0.5:

∫xn dx = xn+1n+1 + C

∫x0.5 dx = x1.51.5 + 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 x33 + 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 ∫(ew − 3) dw ?

Use the Difference Rule:

∫(ew − 3) dw =∫ew dw − ∫3 dw

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

= ew − 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

Final Advice

• Get plenty of practice
• Don't forget the dx (or dz, etc)
• Don't forget the + C

6834, 6835, 6836, 6837, 6838, 6839, 6840, 6841, 6842, 6843 Integration Calculus Index