Table of Contents

## Definition

Integration is used to find areas, volumes, central points and often is used to find the area underneath the graph of a function.

## Common Functions of Integration

| Function | Integral |

Constant | ∫ a dx | ax + c |

Variable | ∫ x dx | |

Square | ∫ x^{2} dx | |

Reciprocal | ∫ (1/x) dx | ln|x| + c |

Exponential | ∫ e^{x} dx | e^{x} + c |

∫ a^{x} dx | ||

∫ 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 of Integration

Rules | Function | Integral |

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

Power Rule (n≠-1) | ∫ x^{n} dx | |

Sum Rule | ∫ (f + g) dx | ∫f dx + ∫g dx |

Difference Rule | ∫ (f – g) dx | ∫f dx – ∫g dx |

Rules | Function | Application | Integral |

Multiplication by constant | ∫ 5 cos(x) dx | 5 ∫ cos(x) dx | 5 sin(x) + c |

Power Rule (n ≠ -1) | ∫ x^{2} dx | ||

Sum Rule | ∫ (cos(x) + x^{2}) dx | ∫ cos(x) dx + ∫ x^{2} dx | |

Difference Rule | ∫ (x^{3} – sin(x)) dx | ∫ x^{3} dx – ∫ sin(x) dx |

### Examples

7.Find the area underneath the curve y = x2 + 3 from x = -1 to x = 1.

Answer:

8. Determine the volume of the solid of the revolution generated when the region is bounded by f(x) = -2×2 + 8 and the x -axis revolved about x-axis.

Answer:

Determine intersect point

-2x^{2} + 8 = 0

2x^{2} = 8

x^{2} = 4

x = 2 or x = -2