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 inverse substitutions.

With inverse substitutions, we let $x = g(\theta)$ , where typically $g$ is a trig function. Then $dx = g'(\theta)\, d\theta$ , and we can always rewrite the integrand in terms of $\theta$ and $d\theta$ . The tricky part is converting everything back to $x$ at the end of the problem.

There are three very useful trig substitution:

For integrals involving $x^2 + a^2$ or $\sqrt{x^2+a^2}$ , use $x = a \tan(\theta)$ .

Then $dx = a\, \sec^2(\theta)\, d\theta$ and $x^2+a^2 = a^2 \sec^2(\theta)$ . To convert back to $x$ , draw a right triangle with opposite side $x$ , adjacent side $a$ and hypotenuse $\sqrt{x^2+a^2}$ and use soh-cah-toa.

For integrals involving $a^2 - x^2$ or $\sqrt{a^2-x^2}$ , use $x=a \sin(\theta)$ .

Then $dx = a \cos(\theta) d\theta$ and $\sqrt{a^2-x^2}= a \cos(\theta)$ . To convert back to $x$ , draw a right triangle with opposite side $x$ , hypotenuse $a$ and adjacent side $\sqrt{a^2-x^2}$ and use soh-cah-toa.

For integrals involving $x^2-a^2$ or $\sqrt{x^2-a^2}$ , use $x=a \sec(\theta)$ .

Then $dx = a \sec(\theta) \tan(\theta)\, d\theta$ and $\sqrt{x^2-a^2}= a \tan(\theta)$ . To convert back to $x$ , draw a right triangle with adjacent side $a$ , hypotenuse $x$ and opposite side $\sqrt{x^2-a^2}$ and use soh-cah-toa.

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 it works