To evaluatewe complete the square for the quadratic expression to obtain then make the substitution simplify and integrate. Example: Complete the square to obtain and The integrand becomes is a standard integral:Neverthelesscan be integrated by multiplyingby the fraction The numerator is the differential of the denominator sinceand To complete the integral we can either substitute […]

## Conic Sections – Circles, Ellipses, Parabolae and Hyperbolae

Circles, ellipses and hyperbolae have similar equations: All these curves are obtained from the intersection of a plane with a cone as shown above. for a circle of ellipses centred at the origin, withfor a circle andfor an ellipse and for a circle or ellipse centred at For a hyperbola, centred at the origin or […]

## Integrals – Reduction Formulae

An integral of the formcannot be integrated in a single step. We must decrease the power ofby integrating by parts, obtaining an integral in which the power ofis smaller by one. We can do this until we are faced with the integralwhich can be easily integrated, obtaining 1. The sequence of integrations may be a […]

## The Length of a Curve

We can find an approximate value for the length of a curve between twovaluesand by approximating the curve by a series of line segments and adding up the length of each line segment. For the curve give above, the length of the line segmentso the length of the curve is approximatelyifandWe can write this asand […]