Integration have the right to be provided to uncover areas, volumes, main points and many valuable things. The is regularly used to find the **area underneath the graph that a role and the x-axis**.

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The an initial rule to recognize is that integrals and also derivatives room opposites!

**Sometimes we can work the end an integral,because we recognize a corresponding derivative.**

### Integration Rules

Here are the most valuable rules, with instances below:

Common FunctionsFunctionIntegralRulesFunctionIntegral

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 | |

Multiplication by constant | ∫cf(x) dx | c∫f(x) dx |

Power preeminence (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 through Parts | See Integration through Parts | |

Substitution Rule | See Integration by Substitution |

### Example: what is the integral the sin(x) ?

**From the table over it is provided as gift −cos(x) + C**

It is written as:

∫sin(x) dx = −cos(x) + C

### Example: what is the integral that 1/x ?

From the table over it is detailed as gift **ln|x| + C**

It is created as:

∫(1/x) dx = ln|x| + C

The vertical bars **||** either next of **x** average absolute value, because we don"t want to give an unfavorable values to the organic logarithm function **ln**.

### Example: What is ∫x3 dx ?

The question is questioning "what is the integral that x3 ?"

We have the right to use the power Rule, whereby n=3:

∫xn dx = *xn+1***n+1** + C

∫x3 dx = *x4***4** + C

### Example: What is ∫√x dx ?

√x is also **x0.5**

We can use the power Rule, wherein n=0.5:

∫xn dx = *xn+1***n+1** + C

∫x0.5 dx = *x1.5***1.5** + C

### Example: What is ∫6x2 dx ?

We can move the 6 exterior the integral:

∫6x2 dx = 6∫x2 dx

And now use the Power dominion on x2:

= 6 *x3***3** + C

Simplify:

= 2x3 + C

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

Use the sum Rule:

∫(cos x + x) dx = ∫cos x dx + ∫x dx

Work the end the integral of each (using table above):

= sin x + x2/2 + C

### Example: What is ∫(ew − 3) dw ?

Use the distinction Rule:

∫(ew − 3) dw =∫ew dw − ∫3 dw

Then work out the integral of every (using table above):

= ew − 3w + C

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### 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

See Integration by Parts

### Substitution Rule

See Integration through Substitution

## Final Advice

Get plenty of practiceDon"t forget the**dx**(or dz, etc)Don"t forget the

**+ C**