Indefinite Integrals

So the integral gives us the area of under a curve and/or the anti-derivative as seen in Intro to Integration. But how do we calculate these things?

### Finding Integrals

There are many many approaches to calculating integrals, but the most basic, which is all that is required in SL, is to look at the derivative table.

 Function Derivative $x^n$ $nx^{n-1}$ $sin(x)$ $cos(x)$ $cos(x)$ $-sin(x)$ $tan(x)$ $\frac{1}{cos^2(x)}$ $e^x$ $e^x$ $ln(x)$ $\frac{1}{x}$

Example 1 Given $g'(x)=x^3$ what is $g(x)$?

Remembering our rule for derivatives of polynomials we know that if $f(x)=x^4$ then $f'(x)=4x^3$

Which is almost what we have, but we don't want the 4. The derivative of $g(x)=\frac{1}{4}x^4$ is equal to $x^3$.

Example 2 Find $\int cos(x) dx$.

Again we know that the derivative of $sin(x)$ is equal to $cos(x)$. So we can say that:

(1)
\begin{align} \int cos(x) dx = sin(x) \end{align}

### Integration Constant

There is one problem, a detail that I have skipped over. The derivative of $x^2+1$ is the same as the derivative of $x^2-3$. The derivative of both functions is $2x$. In fact the derivative of the family of functions $x^2 + c$ is $2x$.

This means there are an infinite number of functions whose derivative is $2x$ so when asked to solve

(2)
\begin{align} \int 2x dx \end{align}

There are an infinite number of answers! So the answer to the integral above is:

(3)
\begin{align} \int 2x dx = x^2+C \end{align}

Where C is called the integration constant. The value of the integration constant can only be determined if you are given more information. This extra information is usually referred to as "initial conditions."

### A few more Examples

(4)
\begin{align} \int sin(x) dx = -cos(x) + C \end{align}
(5)
\begin{align} \int 2 \frac{1}{x} dx = 2 \int \frac{1}{x} dx = 2 ln(x) + C , x>0 \end{align}

Note that constants can be pulled through the integral. And that the integral is only defined for when x is greater than 0.

(6)
\begin{align} \int (2x^3 + 2x^2 + 3) dx = \frac{1}{2}x^4 + \frac{2}{3}x^3 + 3x + C \end{align}

In a polynomial, much like with a derivative, each term can be dealt with individually. In fact the intregral above can be rewritten as:

(7)
\begin{align} \int (2x^3 + 2x^2 + 3) dx =\int 2x^3 dx + \int 2x^2 dx + \int 3 dx \end{align}

### A few last details

There are books filled with tables of integrals. For the most part the integrals you will be required to calculate are fairly simple. In SL you are required or expected to know how to find the integral of any of the functions:

$x^n, e^x, sin(x), cos(x), ln(x)$.

PLUS the composites of any of these with the linear function $ax+b$. Or in other words things that looks like:

(8)
\begin{align} \int sin(3x+2) dx = \frac{-1}{3}cos(3x+2) \end{align}

or

(9)
\begin{align} \int e^{-2x+3}=\frac{-1}{2}e^{-2x+3} \end{align}

These are pretty easy to do and take on the general form:

(10)
\begin{align} \int f'(ax+b) dx = \frac{1}{a} f(ax+b) \end{align}

It could also be handy to know that in general:

(11)
\begin{align} \int a x^n dx = \frac{a}{n+1} x^{n+1} +C \end{align}

Anybody else feel like things got a whole lot more complicated all of a sudden?