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

October 27, 2009 Leave a comment

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