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 | 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=0.5:
∫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