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.
|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|
|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|
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 ∫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=½:
∫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
= 2x3 + C
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
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
= 8∫z dz + 4∫z3 dz − 6∫z2 dz
= 8z2/2 + 4z4/4 − 6z3/3 + C
= 4z2 + z4 − 2z3 + C
Integration by Parts