# Integration Rules

## IntegrationIntegration can be used to find areas, volumes, central points and many useful things. But it is often used to find the |

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**