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Math 2565, Spring 2020

Trigonometric Substitution

January 16, 2020

Sean Fitzpatrick
University of Lethbridge

Warm-Up

Evaluate the following integrals:

  1. \(\displaystyle \int \sin^3(x)\cos^2(x)\,dx\)
  2. \(\displaystyle \int \sin^2(x)\cos^2(x)\,dx\)
  3. \(\displaystyle \int \sec^3(x)\tan(x)\,dx\)

Think about (but don't actually try to solve — yet) the integral

\begin{equation*} \int \sec^3(x)\tan^2(x)\,dx\text{.} \end{equation*}

Quiz

Sometimes a trigonometric or hyperbolic substitution can help with integration. Possible options include:

  • \(\displaystyle x=a\sin\theta\)
  • \(\displaystyle x=a\tan\theta\)
  • \(\displaystyle x=a\sec\theta\)
  • \(\displaystyle x=a\sinh(t)\)
  • \(\displaystyle x=a\cosh(t)\)

For each expression below, indicate which of the above options will work.

  1. \(\displaystyle \sqrt{16-x^2}\)
  2. \(\displaystyle \sqrt{9+x^2}\)
  3. \(\displaystyle \sqrt{x^2-4}\)

Integrals of the form \(\int \tan^m(x)\sec^n(x)\) (Part I)

Case I: \(n\geq 2\) is even.

Use \(u=\tan(x), du = \sec^2(x)\,dx\text{,}\) and the identity \(1+\tan^2(x)=\sec^2(x)\text{.}\)

Examples:

  1. \(\displaystyle \int \tan^4(x)\sec^6(x)\,dx\)
  2. \(\displaystyle \int \tan^5(x)\sec^4(x)\,dx\)

Integrals of the form \(\int \tan^m(x)\sec^n(x)\) (Part II)

Case II: \(m\geq 1\) is odd, and \(n\geq 1\)

Use \(u=\sec(x)\text{,}\) so \(du = \sec(x)\tan(x)\,dx\text{,}\) and the identity \(1+\tan^2(x)=\sec^2(x)\text{.}\)

Examples:

  1. \(\displaystyle \int \tan^3(x)\sec^3(x)\,dx\)
  2. \(\displaystyle \int \tan(x)\sec^4(x)\,dx\)

Integrals of the form \(\int \tan^m(x)\sec^n(x)\) (Part III)

Case III: \(m\) is even and \(n\) is odd — the danger zone. Typically this requires multiple integrations by parts or reduction formulas.

Examples:

  1. \(\displaystyle \int \sec^3(x)\,dx\)
  2. \(\displaystyle \int \tan^2(x)\sec^3(x)\,dx\)

Some sneaky examples

  • \(\displaystyle \int \frac{1}{1+\sin(x)}\,dx\)

  • \(\displaystyle \int \frac{1}{\sqrt{1-\cos(2x)}}\,dx\)

Integrals involving \(a^2-x^2\)

Examples:

  1. \(\di\int \sqrt{9-x^2}\,dx\)

  2. \(\di \int_{-3}^3\sqrt{9-x^2}\,dx\)

  3. \(\di \int\frac{\sqrt{4-x^2}}{x^2}\,dx\)

  4. \(\displaystyle \int_{-1}^1x^2\sqrt{1-x^2}\,dx\)

Integrals involving \(a^2+x^2\)

Examples:

  1. \(\di \int\sqrt{4+x^2}\,dx\)

  2. \(\di\int\frac{x^2}{\sqrt{x^2+3}}\,dx\)

  3. \(\di \int\frac{x}{(x^2+9)^{3/2}}\,dx\)

  4. \(\displaystyle \int\frac{1}{(x^2+4x+13)^2}\,dx\)

Integrals involving \(x^2-a^2\)

Examples:

  1. \(\di \int \sqrt{x^2-16}\,dx\)

  2. \(\di \int_4^8 \sqrt{x^2-16}\,dx\)

  3. \(\di \int \frac{3x^2}{\sqrt{x^2-4}}\,dx\)

  4. \(\displaystyle \int \frac{\sqrt{x^2-1}}{x}\,dx\)