# Integration Short Cuts

Integration has all sorts of useful applications. For example, you might have been given a function definition that tells you how quickly students come up with excuses to avoid doing extra homework and want to work out the total number of excuses that have been made in the past 5 days. Integration will help you out here as it gives you a way to find the area under the 'rate of making excuses' curve between day 0 and day 5:
$\displaystyle{\text{Total Excuses} = \int_0^5 \frac{d\text{Excuse}}{dt} \; dt}$.
This area represents the total number of excuses made over these 5 days.

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 $\displaystyle{\int (f - g)(x) \; dx}$ $\displaystyle{\int f(x) \; dx - \int g(x) \; dx}$
Mathematicians have a special name for the first rule and the last two rules. They refer to these properties as the 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

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

### Example 2: applying the power rule

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

Use the power rule with $n = 4$:

$\displaystyle{ \int x^4\; dx = \dfrac{x^5}{4 + 1} + C = \dfrac{x^5}{5} + C}$.
Watch out! Lots of students get confused here and apply the power rule for derivatives. Try to remember: when you're 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.

$\displaystyle{\int \sqrt{x} \; dx = \int x^{\frac{1}{2}}\; dx = \dfrac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \dfrac{x^{1.5}}{1.5} + C}.$

### Example 4:constant multiplication

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

Bring the $7$ out to the front of the integral:

$\displaystyle{\int 7x^3 \; dx = 7\int x^3\; dx}.$
Now use the power rule on the $x^3$:
$\displaystyle{\int 7x^3 \; dx = 7\int x^3\; dx = 7 \left( \dfrac{x^4}{4} + C\right) = \dfrac{7x^4}{4} + C'},$
where $C' = 7C$.

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

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

$\displaystyle{\int \sec^2 x + x^2 \; dx = \int \sec^2 x\; dx + \int x^2 \; dx}.$
Now use the table of common integrals to work out the integral of each part:
$\displaystyle{\int \sec^2 x + x^2 \; dx = \int \sec^2 x\; dx + \int x^2 \; dx = \tan x + \dfrac{x^3}{3} + C.}$

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

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

$\displaystyle{\int \cos x - e^x \; dx = \int \cos x\; dx - \int e^x \; dx}.$.
Now use the table of common integrals to work out the integral of each bit:
$\displaystyle{\int \cos x - e^x \; dx = \int \cos x\; dx - \int e^x \; dx = \sin x - e^x + C}$.

### 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:

$\displaystyle{\int 5\sin x + 3x^3 - e^x \; dx = \int 5\sin x\; dx + \int 3x^3 \; dx - \int e^x\; dx}$.
Next use the rule about constant multiplication to pull the constants out to the front of the integrals:
$\displaystyle{\int 5\sin x + 3x^3 - e^x \; dx = \int 5\sin x\; dx + \int 3x^3 \; dx - \int e^x\; dx = 5 \int \sin x \; dx + 3 \int x^3 \; dx - \int e^x \; dx}$.
Finally, use the power rule and common integrals to evaluate each part:
$\displaystyle{\int 5\sin x + 3x^3 - e^x \; dx = 5 \int \sin x \; dx + 3 \int x^3 \; dx - \int e^x \; dx = - 5 \cos x + \frac{3x^4}{4} - e^x + C }$.
So, we've exploited the linearity of the integral to calculate:
$\displaystyle{\int 5\sin x + 3x^3 - e^x \; dx = - 5 \cos x + \frac{3x^4}{4} - e^x + C }$.

## 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:

$\displaystyle{F(x) = \int x^3 \; dx = \frac{x^4}{4} + C}$.
Next, plug in $F(2) = 5$ and solve to find the value of $C$:
\begin{align*} F(2) = 5 &= \dfrac{2^4}{4} + C \\ &=\dfrac{16}{4} + C\\ &= 4 + C\\ C &= 1. \end{align*}.
So $F(x) = \dfrac{x^4}{4} + 1$.

### 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.

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