Integration rules

When a function is needed to be integrated, there are certain rules that can be used for this. Most integrals can be calculated using a list of standard integrals. This makes it easy to do integration on most functions.

It is important to notice the + c in every integral. This constant is there because a constant of a function is removed, when a function is derived. This will then give problems if the derivative of the same function is integrated back again, since the original constant in the function will not be there anymore. This results in the wrong function which is not equal to the original function, which must be the case, thus it is important to add the + c when integrating a function.

Standard Integration rules

In the following tables, the standard rules for integrating functions are shown.

Function	Integral
$\int a\;dx$	$ax + c$
$\int ax^n\;dx$	$\frac{a}{n+1}x^{n+1} + c$ where $n \neq -1$
$\int \frac{1}{x}\;dx$	$ln(\\|x\\|) + c$
$\int e^x\;dx$	$e^x + c$
$\int a^x\;dx$	$\frac{a^x}{ln(a)} + c$
$\int ln(x)\;dx$	$xln(x)-x + c$
$\int sin(x)\;dx$	$-cos(x) + c$
$\int cos(x)\;dx$	$sin(x)$

Since the function $ln(x)$ can not take a negative value of x, absolute value signs are used in the integral of the function $\frac{1}{x}$

There also are rules for rewriting integrals to a sometimes easier form. These are listed in the next table.

Integral	Integral of other form
$\int{}{}a\;f(x)\;dx$	$a\int{}{}f(x)\;dx$
$\int{}{}(f(x)+g(x))\;dx$	$\int{}{}f(x)\;dx + \int{}{} g(x)\;dx$

Now a few rules for integration and rewriting integrals are shown. In the next articles a few other methods for solving integrals of functions that can not be solved using these standard integration rules will be shown.

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Standard Integration rules

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