You can always factor the
denominator down into linear (degree one) and quadratic (degree two)
factors. If you have factors that are cubic or higher, you should be
able to factor more.
To factor a cubic, test factors of
the constant term divided by factors of the leading coefficient as
roots. A quick way to check if they work is synthetic substitution
(also known as synthetic division).
Sometimes you can figure out the A
and B by plugging in a particular value of x to
both sides of the equation. (Choose a value that makes one of the
expressions equal to 0.)
If the degree of the numerator is
the same or greater than the degree of the denominator, use long
division first. Based on the results of your long division, split
your integral into a polynomial part (easy to integrate) and a
rational function that you can apply partial fractions to.
If you have a quadratic in the
denominator that doesnt factor, then follow the following steps
= denominator first.
Then separate into
On the first part, use ln |u|.
On the second, complete the
square and use a trigonometric substitution. It should always be a
tangent substitution. (If it is a sine or secant substitution, then
you could have factored the quadratic.)
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.