Trig Substitution

So far we have considered quadratic expressions with no linear $x$ term. In this video we show how to handle expressions like $\sqrt{x^2 + 4x - 5}$ or $x^2 + 2x + 5$ . The key is to complete the square.

Recall that any quadratic polynomial $x^2 +bx +c$ can be converted to a perfect square plus a constant term.

$x^2+bx+c= \left(x + \frac{b}{2}\right)^2+c - \left(\frac{b}{2}\right)^2=\left(x + \frac{b}{2}\right)^2 + \left(c-\frac{b^2}{4}\right)$ This works since when we square

$x+\frac{b}{2}$ we get

$x^2+bx+\left(\frac{b}{2}\right)^2$ and we must subtract the extra term

$\left(\frac{b}{2}\right)^2$ .

Examples of completing the square

$x^2-6x+5=\left(x-\frac{6}{2}\right)^2+5-\left(\frac{6}{2}\right)^2=(x-3)^2+5-9=(x-3)^2-4$
$2-8x-x^2=-(x^2+8x-2)=-\left(\left(x+\frac{8}{2}\right)^2-2-(8/2)^2\right)=-\left(\left(x+\frac{8}{2}\right)^2-2-16\right)=18-(x+4)^2$ Notice: we must have the coefficient of the quadratic term, $x^2$ , equal to one, so we factored out the minus.

DO: Complete the square for $x^2+2x+5$ and $5-4x-x^2$ . These are computed in the video so you can check your work when you watch it.

There are two ways to integrate after completing the square. One is to do a $u$ -substitution first, substituting $u=x+\frac{b}{2}$ , and make the stubstitution. After the $u$ -sub we will have an obvious trig substitution integrand. The second method skips the $u$ -sub, and does the trig substitution on the completed square. The video uses the second method.

Example of $u$ -substitution before doing trig substitution

In the video above, we completed the square to get $\displaystyle\int\frac{dx}{x^2+2x+5}=\int\frac{dx}{(x+1)^2+4}$ We want the form $x^2+a^2$ , or after the substitution, $u^2+a^2$ . Set $u=(x+1)$ . DO: Do this $u$ -sub, and then try to work this through before looking ahead.

$\int\frac{dx}{(x+1)^2+4} \overset{\fbox{$ \,\,u\,=\,\,x+1\\du\,=\,\,dx$}}{=}\int\frac{du}{u^2+4}\overset{\fbox{$\,\, u\,=\,2\tan\theta\\du\,=\,2\sec^2\theta\,d\theta$}}{=}=\int\frac{2\sec^2\theta}{4\tan^2\theta+4}\,d\theta$

$=\int\frac{\sec^2\theta}{2\sec^2\theta}\,d\theta=\int\tfrac12 d\theta=\tfrac12\theta+C=\tfrac12\tan^{-1}\left(\frac{x+1}{2}\right)+C$
We got

$\theta$ in terms of

$x$ by first writing it in terms of

$u$ .

$\theta=\tan^{-1}\left(\frac{u}{2}\right)=\tan^{-1}\left(\frac{x+1}{2}\right).$

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

Completing the Square

Examples of completing the square

Example of uu-substitution before doing trig substitution

Example of $u$ -substitution before doing trig substitution