# Calculus

### Chapters

### Integration Short Cuts

# Integration Short Cuts

Other applications might include finding volumes of solids, areas under more usual graphs like the one in the picture (no you don't really have to cut the area up into little squares and count them), lengths of arcs, velocities of objects whose acceleration functions have been given, distances travelled, and centres of mass. Physicists really know how to exploit a good integral!

So that you don't have to continually calculate integrals using limits of sums of shapes that you can fit under, or over (or around) the curve, there's a list
of integrals of common functions that you can look up. There are also rules that can help you to calculate the antiderivatives of wonkier functions, and a handy little theorem
called the `Fundamental Theorem of Calculus`

that actually lets you evaluate definite integrals, and retrieve useful information like the total number of excuses from
the rate of excuses curve. Why not have a look at the article about definite integrals for more information on this useful topic?

In this article, we'll give you a brief
overview of the rules of integration, as well as the antiderivatives of some common functions. We'll also work through a few examples so that you
can get a better idea of how the common integrals and rules of integration are applied. If you're interested in finding out more, have a look at
the articles on integration by substitution and integration by parts. Oh and, by the way, integration is sometimes viewed as the opposite of differentiation, so
you might see an `indefinite integral`

(one without limits) referred to as an `antiderivative`

in many books.

### Some Common Antiderivatives (Integrals)

Function description | Integral expression | Antiderivative |

Constant Function (\(k\) is for 'konstant') | \(\displaystyle{\int k \; dx}\) | \(kx + C\) |

Linear Variable | \(\displaystyle{\int x \; dx}\) | \(\dfrac{x^2}{2} + C\) |

Square | \(\displaystyle{\int x^2 \; dx}\) | \( \dfrac{x^3}{3} + C\) |

Reciprocal | \(\displaystyle{\int \dfrac{1}{x}\;dx}\) | \( \ln |x| + C\) |

Exponentials | \(\displaystyle{\int e^x \;dx}\) | \( e^x + C\) |

Exponentials | \(\displaystyle{\int a^x \;dx}\) | \( \dfrac{a^x}{\ln (a)} + C\) |

Logarithms | \(\displaystyle{\int \ln (x) \;dx}\) | \( x\ln (x) - x + C\) |

Trig functions (don't forget \(x\) is measured in radians!) | \(\displaystyle{\int \sin x \; dx}\) | \(-\cos x + C\) |

\(\displaystyle{\int \cos x \; dx}\) | \(\sin x + C\) | |

\(\displaystyle{\int \sec^2 x \; dx}\) | \(\tan x + C\) |

### Rules of Integration

And now for the rules that help you stitch the above antiderivatives together to come up with the integrals of some really impressive functions. This is great fun at parties! We'll start with the basics, and deal with the more complicated ones like integration by substitution and integration by parts in later articles.

RuleIntegral expressionTransformed Integral

Multiplication by a constant | \(\displaystyle{\int kf(x) \; dx}\) | \(\displaystyle{k\int f(x) \;dx}\) |

Power Rule (\(n \neq -1\)) | \(\displaystyle{\int x^n \; dx}\) | \(\dfrac{x^{n + 1}}{n + 1} + C\) |

Sum of Integrals | \(\displaystyle{\int (f + g)(x) \; dx}\) | \(\displaystyle{ \int f(x) \; dx + \int g(x) \; dx}\) |

Difference of Integrals | \(\displaystyle{\int (f - g)(x) \; dx}\) | \( \displaystyle{\int f(x) \; dx - \int g(x) \; dx}\) |

`linearity`

of the integral. Basically,
all this means is that integrals behave the way you want them to when it comes to multiplication by products, sums and differences. You can pull
the constant out the front of the integral, and you can split the integral up into bite sized pieces across the plus and minus signs. The property is called `linearity`

because it's the same property that straight line functions have. If \(f(x)\) is a straight line function, then \(f(kx + y) = kf(x) + f(y)\) just like
\(\displaystyle{\int kf(x) + g(x) \; dx = k \int f(x) \; dx + \int g(x) \; dx}\).
Unfortunately, products don't play quite so nicely, but then they weren't so friendly with derivatives either. Techniques such as integration by substitution and integration by parts will help us to deal with the integrals of products of functions.

## Examples of applying the rules

Let's finish by using the rules to find the integrals of some functions. Don't forget that the table of common antiderivatives will be a big help here.

### Example 1

Find the integral of \(\displaystyle{\sec^2 (x)}\).

In the table of common antiderivatives, the antiderivative of \(\sec^2 (x)\) is listed as \(\tan (x) + C\). So

### Example 2: applying the power rule

Evaluate \(\displaystyle{\int x^4 \; dx}\).

Use the power rule with \(n = 4\):

**integrating**, the power of \(x\) goes

**up**. When you're

**differentiating**, the power of \(x\) goes

**down**("d" for "differentiation").

### Example 3: more on the power rule - fractional indices

Evaluate \(\displaystyle{\int \sqrt{x}\; dx}\).

Remember your index laws: \(\sqrt{x} = x^{\frac{1}{2}}\), so we **can** use the power rule here.

### Example 4:constant multiplication

Find \(\displaystyle{\int 7x^3\; dx}\).

Bring the \(7\) out to the front of the integral:

### Example 5: the integral of a sum is the sum of the integrals

Find \(\displaystyle{\int \sec^2 x + x^2 \; dx}.\)

### Example 6: the integral of a difference is the difference of the integrals

Find \(\displaystyle{\int \cos x - e^x\; dx}\).

### Example 7: using all the rules together (exploiting the linearity of the integral)

Find the integral of \(\displaystyle{\int 5\sin x + 3x^3 - e^x \; dx}\).

Use the sum and difference rules to split the integral into three parts:

## What about all those constants?

You'll have noticed that we've added a constant to the end of each integral we've calculated. If you've done integration at school,
your teacher will have insisted that you add constants to indefinite integrals. For each indefinite integral, we've found a **family**
of functions that could be differentiated to give the `integrand`

(the thing we're integrating). The effect of different values of
the constant is to move the graph of the function (and all its values) up and down the \(xy\)-plane. Sometimes a problem will give you a condition
that the antiderivative must satisfy, such as \(F(0) = 0\). You can plug this condition into the formula for the anti-derivative to
work out what the constant is.

### Example:

Find the function \(\displaystyle{F(x) = \int x^3 \;dx}\) that satisfies \(F(2) = 5\).

First find the indefinite integral using the power rule:

### Description

Calculus is the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are *differential calculus* and *integral calculus*.

### Environment

It is considered a good practice to take notes and revise what you learnt and practice it.

### Audience

Grade 9+ Students

### Learning Objectives

Familiarize yourself with Calculus topics such as Limits, Functions, Differentiability etc

Author: Subject Coach

Added on: 23rd Nov 2017

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