---
title: 8.4 Trig Substitution (Module 9)
tags: [Notebooks/MA 242/Notes]
created: '2021-03-01T18:38:05.668Z'
modified: '2021-03-24T17:58:23.165Z'
---

# 8.4 Trig Substitution (Module 9)
Integrals that involve &sqrt(a^2 -x^2)& or &sqrt(a^2 + x^2)&
Use right triangle trigonometry
Use trig functions &sin theta&, &tan theta&, and &sec theta&
Think &x/a& using Pythagorean Theorem

```
   /|
c / | b
 /  |
/   |
  a
  ```
  &&c = sqrt(a^2 + b^2)&&
  &&a = sqrt(c^2 - b^2)&&
  &&b = sqrt(c^2 - a^2)&&
  &&sin theta = "O"/"H"&&
  &&tan theta = "O"/"A"&&
  &&sec theta = "H"/"A"&&

  (Basically: we want to convert one of the above "square root of weird things" into trig functions that are easier to integrate, integrate them, and then undo all our wacky replacements.)

#### Example
&&sqrt(a^2 - x^2)&&

&&a -> "Hypotenuese"&&
&&x -> "Opposite"&&
*N.B: You want to produce &x/a& in your equations, as &a/x& is a pain to work with*

We can now create a triangle with a hypotenuese of &a&, an opposite of &x&, and an adjacent of &sqrt(a^2-x^2)&.
We now want to solve for x, dx, theta and the radical...

&&sin theta = x/a&&
&&a sin theta = x&&
&&a cos theta d theta= dx&&
&&theta = sin^(-1) (x/a)&&

We now want to rewrite our radical in terms of theta, not x:
&&sqrt(a^2 - (asin theta)^2)&&
&&sqrt(a^2 - a^2 sin^2 theta)&&
&&a sqrt(1 - sin^2 theta)&&
&&a sqrt(cos^2 theta)&&
&&a cos theta = sqrt(a^2-x^2)&&

#### An Actual Example
&&int (x^2)/(sqrt(9-x^2))dx&&
&&a = 3&&

&&sin theta = x/3&&
&&x = 3 sin theta&&
&&dx = 3cos theta d theta&&

Substitute in x into the radical
&&sqrt(9-x^2)&&
&&sqrt(9 - (3 sin theta)^2)&&
&&sqrt(9 - 9 sin^2 theta)&&
&&3sqrt(1-sin^2 theta)&&
&&3sqrt(cos^2 theta)&&
&&3cos theta&&
&&therefore sqrt(9-x^2) = 3cos theta&&

Find theta
&&sin theta = x/3&&
&&theta = arcsin(x/3)&&

Rewrite the original integral with our funny numbers:
**DO NOT FORGET THE &dx&!!**
&&int x^2/sqrt(9-x^2) dx&&
&&int (9 sin^2 theta)/(3cos theta) 3 cos theta d theta&&
&&9 int sin^2 theta / cos theta * cos theta d theta&&
&&9 int sin^2 theta d theta&&
&&9 int (1-cos 2 theta)/2 d theta&&
&&9/2 int (1 - cos 2theta) d theta&&
&&9/2 (theta - (sin 2 theta)/2) +C&&
Power unreduction formulas!!
&&9/2 (theta - 2 sin theta cos theta) +C&&
&&9/2(arcsin(x/3) - xsqrt(9-x^2)/9)&&

When working with arcsin and arctangent, certain restrictions apply:
&&-pi/2 <= theta <= pi /2&&

#### Another Example
&&int x^3/sqrt(x^2 +4 dx)&&

Hyp: &sqrt(x^2 + 4)&
Opp: &x&
Adj: &2&
&&tan theta = x/2&&
&&x = 2 tan theta&&
&&dx = 2 sec^2 theta d theta&&
&&sqrt(x^2 + 4 )dx&&
&&sqrt(4 tan^2 theta + 4)&&
&&2 sec theta&&
&&int (8 tan^3 theta)/(2 sec theta) 2sec^2 theta d theta&&
&&int 8tan^3 theta sec theta d theta&&
&&8 int (tan^2 theta)sec theta tan theta d theta&&
&&8 int (sec^2 theta-1)sec theta tan theta d theta&&
&&u = sec theta&&
&&du = sec theta tan theta d theta&&
&&8 int (u^2 -1) du&&
&&8 (u^3/3 - u)&&
&&8 (sec^3 theta/3 - sec theta)&&
&&because&&
&&x = 2 tan theta&&
&&sec theta = sqrt(x^2+4)/2&&
&&1/3 (x^2 +4)^(3/2) - 4sqrt(x^2+4) +C&&

#### EGG SAMPLE
&&sqrt(x^2 - a^2)&&
hypotenuese: &x&, adjacent: &a&, opposite: &sqrt(x^2-a^2)&
&&sec theta = x/a&&
&&x = a sec theta&&
&&dx = a sec theta tan theta d theta&&
&&sqrt(a^2 sec^2 theta - a^2)&&
&&asqrt(sec^2 theta - 1)&&
&&a tan theta&&

Important:
&&theta = sec^-1(x/a)&&
&&0 <= theta <= pi/2 if x/a >= 1&&
&&pi/2 <= theta <= oi if x/a <= -1&&

&&int dx/sqrt(25x^2-4)&&
&&"Adj: " a=2&&
&&"Hyp: " 5x&&
&&"Opp: " sqrt(25x^2-4)&&

&&sec theta = (5x)/2&&
&&x = 2/5 sec theta&&
&&dx = 2/5 sec theta tan theta d theta&&

&&int sqrt(25(2/5 sec theta)^2 - 4) 2/5 sec theta tan theta d theta&&
&&2/5 int sqrt(4 sec^2 theta - 4) sec theta tan theta d theta&&
&&1/5 int 1/sqrt(sec^2 theta -1) sec theta tan theta d theta&&
&&1/5 int 1/tan theta sec theta tan theta d theta&&
&&1/5 int sec theta d theta&&
&&1/5 ln|sec theta + tan theta| + C&&
&&1/5 ln | (5x)/2 + sqrt(25x^2 -4)/2|&&

#### Another Example
&&int_-2^2 dx/(4+x^2)&&
&&int_-2^2 (1/sqrt(4+x^2))^2 dx&&
&&a = 2&&
&&tan theta = x/2&&
&&x = 2 tan theta&&
&&dx = 2 sec^2 theta d theta&&

&&sqrt(4+x^2) = 2 sec theta&&
&&4 + x^2 = 4 sec^2 theta&&
&&theta = arctan(x/2)&&

#### Example 2a
&&int_sqrt(2)^2 1/(t^2 sqrt(t^2 -1)) dt&&
&&t = sec theta&&
&&dt = sec theta tan theta d theta&&
&&theta = arcsec t&&

&&int_(pi/4)^(pi/3) cos theta d theta&&
&&sin theta|_(pi/4)^(pi/3) = (sqrt(3) - sqrt(2))/2&&