Integrating Fractions – using the natrual logarithm

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

October 27th, 2009 admin Leave a comment Go to comments

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) $\int \! \frac{x}{x^2 + 5} \, dx$ , 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 $\frac{1}{2} \int \! \frac{2x}{x^2 + 5} \, dx = \frac{1}{2}ln|x^2 + 5| + c$

Trevor Pythagoras Maths