1. Collect homework.
2. Recall:
3. Given two sides of a right triangle, a and u,
what are the three possibilities that exist?
Draw the triangles and label the third side in each.
We will use these three triangles any time that we have
an integral involving the sum or difference of two
squares. We will use the letter a to represent
constants and the letter u to represent variables.
This requires no memorization, since you just set up
the triangle so that the third side corresponds to
what you are looking for (a
2 - u
2, u
2 - a
2, or a
2 + u
2).
Here is how you use the triangle:
What we are doing is taking an integral involving
the sum or the difference of two squares which we
can not integrate (because it is in the denominator or
because it is under a square root), and making an
appropriate trigonometric substitution so that we obtain
an integral of just one term. We can then integrate it
or take its square root.
4. Examples: