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Math 2565, Spring 2020

Trigonometric Integrals

January 14, 2020

Sean Fitzpatrick
University of Lethbridge

Quiz

  1. Using integration by parts, evaluate

    \begin{equation*} \int \ln(x)\,dx\text{.} \end{equation*}

  2. For integrals of the form \(\displaystyle \int \sin^m(x)\cos^n(x)\,dx\text{,}\) which case is hardest (in that it requires the most work)?

    1. \(m\) and \(n\) are both odd.

    2. \(m\) and \(n\) are both even.

    3. One of \(m\) or \(n\) is even, and the other is odd.

Integration by parts (recap)

Here are a few more problems to try:

  • \(\displaystyle \int \arcsin(x)\,dx\)
  • \(\displaystyle \int x^3\ln(x)\,dx\)
  • \(\displaystyle \int \ln(x^2-1)\,dx\)
  • \(\displaystyle \int e^{\sqrt{x}}\,dx\)

Any others (perhaps from the homework) you'd like to see?

Trig integrals — \(\int \sin^m(x)\cos^n(x)\,dx\)

As long as at least one power is odd, use substitution and Pythagorean identity.

Examples:

  1. \(\displaystyle \int \sin^4(x)\cos^3(x)\,dx\)
  2. \(\displaystyle \int \sin^3(x)\cos^5(x)\,dx\)

When there are only even powers, we need to use power reduction formulas.

Examples:

  1. \(\displaystyle \int \sin^4(x)\,dx\)
  2. \(\displaystyle \int \sin^2(x)\cos^4(x)\,dx\)

An example with options

Evaluate the integral

\begin{equation*} \int \sin(x)\sin(2x)\,dx \end{equation*}
using:

  • A double-angle identity.

  • A product-to-sum identity.

  • Integration by parts.

Product-to-sum identities

These are derived from the angle addition formulas for sine and cosine. For example:

\begin{align*} \cos(mx+nx) \amp = \cos(mx)\cos(nx)-\sin(mx)\sin(nx)\\ \cos(mx-nx) \amp = \cos(mx)\cos(nx)+\sin(mx)\sin(nx)\text{.} \end{align*}
Adding these gives an identity for \(\cos(mx)\cos(nx)\text{.}\) Subtracting does the same for \(\sin(mx)\sin(nx)\text{.}\)

For products of the from \(\sin(mx)\cos(nx)\text{,}\) use

\begin{equation*} \sin(mx\pm nx) = \sin(mx)\cos(nx)\pm \sin(nx)\cos(mx)\text{.} \end{equation*}
Example: evaluate
\begin{equation*} \int \sin(5x)\cos(9x)\,dx\text{.} \end{equation*}

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

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

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