# Integration Rules

- Integration
- Examples
- Example: what is the integral of sin(x) ?
- Example: what is the integral of 1/x ?
- Power Rule
- Example: What is ∫x3 dx ?
- Example: What is ∫√x dx ?
- Multiplication by constant
- Example: What is ∫6x2 dx ?
- Sum Rule
- Example: What is ∫(cos x + x) dx ?
- Difference Rule
- Example: What is ∫(ew − 3) dw ?
- Sum, Difference, Constant Multiplication And Power Rules
- Example: What is ∫(8z + 4z3 − 6z2) dz ?
- Integration by Parts
- Substitution Rule

- Final Advice

## Integration

consolidation can be used to find areas, volumes, central points and many utilitarian things. It is much used to find the **area underneath the graph of a function and the x-axis** .

The first dominion to know is that integrals and derivatives are opposites !

sometimes we can work out an built-in,

because we know a duplicate derived function.

Reading: Integration Rules

### Integration Rules

here are the most utilitarian 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+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 board 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 erect bars **||** either side of **x** mean absolute respect, because we do n’t want to give negative values to the natural logarithm function **ln** .

### Power Rule

### Example: What is ∫x3 dx ?

The interview is asking “ what is the integral of x3 ? ”

We can use the Power Rule, where n=3 :

∫xn dx = xn+1 **n+1** + C

∫x3 dx = x4 **4** + C

### Example: What is ∫√x dx ?

√x is besides **x0.5**

We can use the Power Rule, where n=0.5 :

∫xn dx = xn+1 **n+1** + C

∫x0.5 dx = x1.5 **1.5** + C

### Multiplication by constant

### Example: What is ∫6×2 dx ?

We can move the 6 outside the integral :

∫6×2 dx = 6∫x2 dx

And immediately use the Power Rule on x2 :

= 6 x3 **3** + C

simplify :

= 2×3 + C

### Sum Rule

### Example: What is ∫(cos x + x) dx ?

Use the Sum convention :

∫ ( cobalt x + x ) dx = ∫cos x dx + ∫x dx

study out the integral of each ( using mesa above ) :

= sine x + x2/2 + C

### Difference Rule

### Example: What is ∫(ew − 3) dw ?

Use the Difference rule :

∫ ( electronic warfare − 3 ) dw =∫ew dw − ∫3 dw

then work out the integral of each ( using table above ) :

= electronic warfare − 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

exponent rule :

= 8z2/2 + 4z4/4 − 6z3/3 + C

simplify :

= 4z2 + z4 − 2z3 + C

### Integration by Parts

See integration by Parts

### Substitution Rule

See consolidation by Substitution

Read more: Is It Hard to Learn Computer Programming?

## Final Advice

- Get plenty of practice
- Don’t forget the
**dx**(or dz, etc) - Don’t forget the
**+ C**

6834, 6835, 6836, 6837, 6838, 6839, 6840, 6841, 6842, 6843