Calculus Methods
27 Trig Sub
This is a method of integration when simple
antidifferentiation (method 14), u-substitution (method 16),
and integration by parts (method 26) do not work. The key is
looking for a polynomial comprising only a squared term and
a constant term.
1)
If the function looks like 
, factor out
so that the
constant we are dealing with is now 1.
2) Identify the type of trigonometric substitution is necessary:
If you forget which substitution to make, remember your
trigonometric Pythagorean identities!
Note the similarities of these identities to the proposed
substitutions above.
3)
Make the substitution, including calculating 
.
4) Integrate.
5) Back substitute. It may be helpful to draw a triangle!
Example
#1: Integrate 
.
1) The coefficient is already 1.
2)
3)
4)
5)
Now we need to know what 
and 
are in terms of 
.
To
do this, we'll draw a triangle. We already know from our
substitution that
. So on our right triangle, we will
call the
angle of the triangle 
, the
adjacent leg 
, and the hypotenuse 
.
By the Pythagorean Theorem, the
remaining leg is thus 
.
 |
|
From
our picture, now, it is easy to see that 
and that
. Our final answer,
then, is
Example
#2: Integrate 
.
1)
2)
3)
4)
5) Here's our triangle:
So
.
On to Method 28 - L'Hopital's Rule
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