Trigonometric substitution ("trig substitution") reduces certain integrals to integrals of trig functions. The idea is to match the given integral against one of the following trig identities:
(a) If the integral contains an expression of the form , try a substitution based on the first identity:
.
(b) If the integral contains an expression of the form , try a substitution based on the second identity:
.
(c) If the integral contains an expression of the form , try a substitution based on the third identity:
.
If you don't obtain one of the identities above after substituting, you've probably used the wrong substitution.
Example. Compute .
The expression " " leads me to try
Plug in:
To put the x back, I draw a right triangle. The substitution was
Since sine is the opposite side divided by the hypotenuse, I get:
I found the adjacent side using
Pythagoras. From the triangle, I see that
. Plugging this back in,
I get
Example. Compute .
To "match" the "4" in " ", I use
(since
). Differentiation gives
, so
. I plug in and simplify:
I have an even power of cosine, so I need to use the double angle
formula for :
I need to put the x's back.
For the terms and
, I need to express everything in
terms of trig functions of
(as opposed to
or
). I use the double angle formulas for sine:
Therefore,
The " " in the first term is not inside a
trig function. For that term,
gives
, so
.
For the second and third terms, draw a right triangle which shows the substitution.
The triangle shows --- the
opposite side is x and the hypotenuse is 2 --- and by Pythagoras the
third side is
. Therefore,
Plugging all of this into the last expression, I have
Example. Compute .
looks like
, so let
. Then
, so
Example. Compute .
This could be done using . But it's easier
to do a u-substitution:
Example. Compute .
looks like
, so
let
. Then
, and
Copyright 2019 by Bruce Ikenaga