Calculus- Early Transcendentals, 2021a

7.3. Trigonometric Substitutions 267
This type of substitution is usually indicated when the function you wish to integrate contains a polynomial
expression that might allow you to use the fundamental identity sin 2 x + cos 2 x = 1 in one of three
forms:
cos 2 x = 1 − sin 2 x sec 2 x = 1 + tan 2 x tan 2 x = sec 2 x − 1.
If your function contains 1 − x 2 , as in the example above, try x = sinu; if it contains 1 + x 2 try x = tanu;
and if it contains x 2 − 1, try x = secu. Sometimes you will need to try something a bit different to handle
constants other than one which we will describe below. First we discuss inverse substitutions.
In a traditional substitution we let u = u(x), i.e., our new variable is defined in terms of x. In an
inverse substitution we let x = g(u), i.e., we assume x can be written in terms of u. We cannot do this
arbitrarily since we do NOT get to “choose” x. For example, an inverse substitution of x = 1willgivean
obviously wrong answer. However, when x = g(u) is an invertible function, then we are really doing a
u-substitution with u = g −1 (x). Now the substitution rule applies.
Sometimes with inverse substitutions involving trig functions we use θ instead of u. Thus, we would
take x = sinθ instead of x = sinu. However, as we discussed above, we would like our inverse substitution
x = g(u) to be a one-to-one function, and x = sinu is not one-to-one. We can overcome this issue by using
the restricted trigonometric functions. The three common trigonometric substitutions are the restricted
sine, restricted tangent and restricted secant. Thus, for sine we use the domain [−π/2, π/2] and for
tangent we use (−π/2, π/2). Depending on the convention chosen, the restricted secant function is
usually defined in one of two ways.
♣
One convention is to restrict secant to the region [0, π/2) ∪ (π/2,π] as shown in the middle graph.
The other convention is to use [0, π/2) ∪ [π, 3π/2) as shown in the right graph. Both choices give a
one-to-one restricted secant function and no universal convention has been adopted. To make the analysis
in this section less cumbersome, we will use the domain [0, π/2) ∪ [π, 3π/2) for the restricted secant
function. Then sec −1 x is defined to be the inverse of this restricted secant function.
Typically trigonometric substitutions are used for problems that involve radical expressions. The table
below outlines when each substitution is typically used along with their intervals of validity.