### Archive

Posts Tagged ‘logarithm’

## Integrating Fractions – using the natrual logarithm – Example tan(x)

From result found be differentiating the natural logarithm,
$\frac{d}{dx} (ln(f(x))) = \frac{f'(x)}{f(x)}$
for some function f(x),

and the fundamental theorem of calculus we cay say that

$\int \! \frac{f'(x)}{f(x)} \, dx = ln|f(x)| + c$ where c is the integration constant

### Simple Example

The most basic example of this is the integration of 1/x,

$\int \! \frac{1}{x} \, dx = ln|x| + c$

### More complex example: Integration of tan(x)

A slightly more complicated example of this is the integration of tan(x). To do this we must remember that $tan(x) = \frac{sin(x)}{cos(x)}$ and notice that $\frac{d}{dx}(cos(x)) = -sin(x)$. This means that -tan(x) is of the form $\frac{f'(x)}{f(x)}$ as required. Using this we can get

$\int \! tan(x) \, dx = \int \! \frac{sin(x)}{cos(x)} \, dx = lan|cos(x)| + c$

### Trick for using this identity

Sometimes we get integrals that are almost in this form but not exactly, eg) $\int \! \frac{x}{5 + x^2} \, dx$, however to solve these we can often factorise a constant so that it is in the required form. In this example we can take out a 2 so we get $\frac{1}{2} \int \! \frac{2x}{ 5 + x^2} \, dx = ln|5 + x^2| + c$