## Content

### Antiderivatives and indefinite integrals

In order to calculate integrals, we now see that it's important to be able to find antiderivatives of functions. An antiderivative of a function \(f(x)\) is a function whose derivative is equal to \(f(x)\). That is, if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\).

Importantly, antiderivatives are not unique. A given function can have many antiderivatives. For instance, the following functions are all antiderivatives of \(x^2\):

\[ \frac{x^3}{3}, \qquad \frac{x^3}{3} + 1, \qquad \frac{x^3}{3} - 42, \qquad \frac{x^3}{3} + \pi. \]However, any two antiderivatives of a given function differ by a constant. This allows us to write a general formula for any antiderivative of \(x^2\):

\[ \frac{x^3}{3} + c, \qquad \text{where \(c\) is a constant}. \]Since, as we saw in the previous section, antiderivatives are intimately connected with areas, we write this antiderivative as

\[ \int x^2 \; dx = \frac{x^3}{3} + c \]and call this an indefinite integral.

An **indefinite integral** is an integral written without terminals; it simply asks us to find a general antiderivative of the integrand. It is not one function but a family of functions, differing by constants; and so the answer must have a '\(+\) constant' term to indicate all antiderivatives.

A definite integral, on the other hand, is an integral with terminals. It is a number which measures an area under a graph between the terminals.

##### Exercise 6

Find the derivatives of the following functions:

- for \(n, c\) any real constants with \(n \neq -1\), \[ f(x) = \frac{1}{n+1} x^{n+1} + c \]
- for \(a,b,c,n\) any real constants with \(a \neq 0\) and \(n \neq -1\), \[ f(x) = \frac{1}{a(n+1)} (ax+b)^{n+1} + c. \]

From our knowledge of differentiation, and in particular the above exercise, we have the following formulas for antiderivatives.

Function | General antiderivative | Comment |
---|---|---|

\(x^n\) | \(\dfrac{1}{n+1} x^{n+1} + c\) | for \(n,c\) any real constants with \(n \neq -1\) |

\((ax+b)^n\) | \(\dfrac{1}{a(n+1)} (ax+b)^{n+1} + c\) | for \(a,b,c,n\) any real constants with \(a \neq 0\), \(n \neq -1\) |

#### Example

Find

- \(\displaystyle\int 1 \; dx\)
- \(\displaystyle\int (3x-1)^4 \; dx\).

#### Solution

- We can observe directly that \(x\) is an antiderivative of 1, or we can use the above rule for the antiderivative of \(x^n\) when \(n=0\). Either way, the general antiderivative is \(x+c\) and so \(\int 1 \; dx = x + c\), where \(c\) is a constant.
- Using the above rule for the antiderivative of \((ax+b)^n\) with \(a=3\), \(b=-1\) and \(n=4\), we obtain \[ \int (3x-1)^4 \; dx = \frac{1}{15} (3x-1)^5 + c, \qquad \text{where \(c\) is a constant.} \]

##### Exercise 7

Find the indefinite integrals

- \(\displaystyle\int \frac{1}{\sqrt{3-2x}} \; dx\)
- \(\displaystyle\int \frac{1}{(3-2x)^2} \; dx\).

The following theorem gives some useful rules for computing integrals.

###### Theorem (Linearity of integration)

- If \(f\) and \(g\) are continuous functions, then \[ \int \big(f(x) \pm g(x)\big) \; dx = \int f(x) \; dx\, \pm \int g(x) \; dx. \]
- If \(f\) is a continuous function and \(k\) is a real constant, then \[ \int k f(x) \; dx = k \int f(x) \; dx. \]

This theorem allows us to compute an antiderivative by treating a function term-by-term and factoring out constants, as the next example illustrates.

#### Example

Find

\[ \int (3x^3 - 4x^2 + 2) \; dx. \]#### Solution

Using the above rules, we can rewrite the integral as follows:

\begin{align*} \int (3x^3 - 4x^2 + 2) \; dx &= \int 3x^3 \; dx \, - \int 4x^2 \; dx \, + \int 2 \; dx \\ &= 3 \int x^3 \; dx \, - 4 \int x^2 \; dx \, + 2 \int 1 \; dx \\ &= \frac{3x^4}{4} - \frac{4x^3}{3} + 2x + c. \end{align*}Note. You might think that, since there are several indefinite integrals on the second line of these equations, there should be several different constants in the answer. But since the sum of several constants is still a constant, we can just write a single constant.

##### Exercise 9

Prove the theorem above (linearity of integration) using similar rules for differentiation.

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