Integration Rules
IntegrationIntegration 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 | x^{2}/2 + C |
Square | ∫x^{2} dx | x^{3}/3 + C |
Reciprocal | ∫(1/x) dx | ln|x| + C |
Exponential | ∫e^{x} dx | e^{x} + C |
∫a^{x} dx | a^{x}/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 | |
∫sec^{2}(x) dx | tan(x) + C | |
Rules | Function |
Integral |
Multiplication by constant | ∫cf(x) dx | c∫f(x) dx |
Power Rule (n≠-1) | ∫x^{n} dx | \frac{x^{n+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
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 ∫x^{3} dx ?
The question is asking "what is the integral of x^{3 }?"
We can use the Power Rule, where n=3:
∫x^{n} dx = \frac{x^{n+1}}{n+1} + C
∫x^{3 }dx = \frac{x^{4}}{4} + C
Example: What is ∫√x dx ?
√x is also x^{0.5}
We can use the Power Rule, where n=½:
∫x^{n} dx = \frac{x^{n+1}}{n+1} + C
∫x^{0.5} dx = \frac{x^{1.5}}{1.5} + C
Multiplication by constant
Example: What is ∫6x^{2} dx ?
We can move the 6 outside the integral:
∫6x^{2} dx = 6∫x^{2} dx
And now use the Power Rule on x^{2}:
= 6 \frac{x^{3}}{3} + C
Simplify:
= 2x^{3} + 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 + x^{2}/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 + 4z^{3} − 6z^{2} dz ?
Use the Sum and Difference Rule:
∫8z + 4z^{3} − 6z^{2} dz =∫8z dz + ∫4z^{3} dz − ∫6z^{2} dz
Constant Multiplication:
= 8∫z dz + 4∫z^{3} dz − 6∫z^{2} dz
Power Rule:
= 8z^{2}/2 + 4z^{4}/4 − 6z^{3}/3 + C
Simplify:
= 4z^{2} + z^{4} − 2z^{3} + C
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