Trig Substitution

Regular $u$ -substitution works by setting $u=g(x)$ for some function $g$ . We have to pick $u$ so that the integrand becomes a nice function of $u$ times $du$ . In the video, we explore what is called inverse substitution, which we use for trig substitution. This is where we substitute something in for $x$ , instead of creating a $u$ that is a function of $x$ .

Trig substitution is appropriate when we have an integrand containing the sum or difference of the squares of a constant and a variable, i.e. one of the forms $x^2+a^2, a^2-x^2$ , and $x^2-a^2$ . Examples integrals for which we would use trig substitution include those below. Notice that regular substitution will not work with these integrals. Also notice it is fine if the $x^2$ has a coefficient, and in the last example, we must complete the square in order to get one of our forms. You should begin to recognize such integrals as trig substitution integrals.

$\int\frac{\sqrt{9-x^2}}{x^2}\,dx,\qquad \int\frac{1}{x^2\sqrt{x^2+4}}\,dx\qquad\int\frac{dx}{(4x^2-25)^{3/2}},\quad\text{ and }\quad \int\frac{x^2}{(3+4x-4x^2)^{3/2}}\,dx$ In trig substitution, we let

$x = g(\theta)$ , where

$g$ is a trig function, and then

$dx = g'(\theta)\, d\theta$ . Since

$x$ and

$dx$ appear in the integrand, we can always rewrite the integrand in terms of

$\theta$ and

$d\theta$ . The question is whether the substitution helps us integrate. Fortunately, we can teach you how to make good substitutions. It is also non-trivial to convert everything back to

$x$ at the end of the problem, or in the case of a definite integral, to change your limits of integration to be in terms of

$\theta$ .

The above three forms indicate the trig subsitutions we will use, and they are easy to remember since you know the derivatives of $\sin^{-1}x,\tan^{-1}x$ , and (maybe) $\sec^{-1}x$ . These trig substitutions are derived in the following video, and summarized on the next page.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

How Trig Substitution Works