Trigonometric Substituion

Due to how triangles and circles interact with trigonometric functions, there are a number of substitutions we can make using trig functions that are helpful. There are three main substitutions we will discuss that are helpful.

  1. \sqrt{a^2-x^2} = \sqrt{a^2 - (a*sin(\theta))^2}, when -\frac{\pi}{2} \le \theta \ge \frac{\pi}{2}
  2. \sqrt{a^2+x^2} = \sqrt{a^2+(a*tan(\theta))^2}, when - \frac{\pi}{2} < \theta < \frac{\pi}{2}
  3. \sqrt{x^2-a^2} = \sqrt{(a*sec(\theta))^2 - a^2}, when 0 \le \theta < \frac{pi}{2} or \pi \le \theta < \frac{3\pi}{2}

Example: Evaluate \int \frac{\sqrt{9-x^2}}{x^2} dx

We can let x = 3sin(\theta), meaning that dx = 3cos(\theta) d\theta, and \sqrt{9 - x^2} = \sqrt{9-9sin^2(\theta) = \sqrt{9cos^2(\theta)} = 3 cos(\theta)

From here, our integral becomes easy to solve.

\int \frac{\sqrt{9-x^2}}{x^2} dx = \int \frac{3cos(\theta)}{9 sin^2(\theta)} 3cos(\theta) d\theta
\int \frac{cos^2(\theta)}{sin^2(\theta)} d\theta = \int cot^2(\theta) d\theta = -cot(\theta) - \theta + C

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