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
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