Integrating any quotient of the formcan be done by making the substitutionFor example to findsubso the integral becomes(1) Complete the integral by substitutingto obtain There is a method that is often quicker. The method is to take out factors to obtain an expressionwhich integrates to give Example: Find Example: Find

## Interval of Convergence of Partial Fractions

We can take the radius of convergence of the fraction to be– there are actually lots of radii of this sort, each corresponding to a different series expansion that represents the fraction– but that is for another time. Complications arise when we have the sum of two or more partial fraction since all the partial […]

## The Logistic Equation

The equation takes a certain form and is used in many models, including the spread of diseases, discussed here. It is based on two assumptions Since each infected person is capable of passing on the disease, the rate of spread of the disease is proportional to the infected population Since the disease may only be […]

## Partial Fractions Rules

An algebraic fraction is any expression of the formwhere andare sums or products of polynomials or both. An expression of this sort typically needs to be written in terms of it’s partial fractions – whereis written as a sum of algebraic fractions – so that it can be integrated. There are rules which determine which […]

## Finding the Line of Intersection of Two Planes

As shown in the diagram above, two planes intersect in a line. We can find the equation of the line by solving the equations of the planes simultaneously, with one extra complication – we have to introduce a parameter. If we take the parameter at being one of the coordinates, this usually simplifies the algebra. […]

## Finding The Perpendicular Line to a Plane Passing Through a Given Point

If a vectoris perpendicular to a planethen it is perpendicular to every vectorand line drawn in the plane, henceWe can find an expression forby subtracting one point in the plane from another. This is shown on the diagram above aswhereis an arbitrary point andis given, so by findingor we find the equation of the plane. […]

## Changing Between Cartesian and Vector Forms of Equations of Lines in Three Dimensions

You may be used to seeing the cartesian –– form of a line as something likeor but expressions like these are not possible in three dimensions. We go right back to the vector form, and start by identifying theandcomponents: HenceFor each of these three components we make the parameter the subject: Each of these expressions […]

## The General Binomial Theorem

To expand the expressionif n is a non negative integer, we can use Pascal’s Triangle or the formula formula for the binomial expansion, We can only use the formula above ifis a non negative integer. Ifis negative or a fraction we can still expand the expression, but must use the formula for the general binomial […]

## Integrating when x and y are in Parametric Form

The area under a curvebetween the limitsandis given by the integral If however,andare given as functions of a parametera lot more work may be required. Instead of integrating with respect towe may integrate with respect to t usinghence Example: A curve is given by the parametric equationsFind the area under the curve between the values […]

## Differential Equations – Separating Variables

It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables – the technique illustrated here. In general a differential equation may haveandterms on both sides, but if the equation […]