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Calculus I: Activities

Mark A. Fitch

Section 5.3 Evaluating Integrals of Inverse Trigonmetric Functions

This section presents materials that explain or enable or use the following standards.
  • Calculate trigonometric derivatives
  • Analyze a function using derivatives
  • Integrate polynomial, trig, and/or exponential functions
First we will consider how we can define inverses of trigonometric functions. Then we will use the same technique we used to discover the derivative of \(ln(x)\) to discover derivatives of these inverses of trigonometric functions. These derivatives provide additional anti-derivatives for integration.

Subsection 5.3.1 Inverses of Trigonometric Functions

First we ask if trigonometric functions have inverses that are functions.

Activity 5.3.1. Checking inverses of trig functions.

Recall that in Activity 5.2.2 we discovered that the inverse of a function is a function when it is strictly increasing or decreasing. Use this to answer the following.
(a)
Consider \(y=sin(x)\text{.}\)
(i)
Graph the sine function.
(ii)
Is the sine function strictly increasing or decreasing?
(iii)
Is the inverse of the sine function a function?
(b)
Consider \(y=cos(x)\text{.}\)
(i)
Graph the cosine function.
(ii)
Is the cosine function strictly increasing or decreasing?
(iii)
Is the inverse of the cosine function a function?
(c)
Consider \(y=tan(x)\text{.}\)
(i)
Graph the tangent function.
(ii)
Is the tangent function strictly increasing or decreasing?
(iii)
Is the inverse of the tangent function a function?

Activity 5.3.2. Domains for Inverse Trigonometric Functions.

We can invert parts of trigonometric functions. We must limit the domain of the function to make this a function.
(a)
Using Figure 5.3.1 adjust the left/right ends of the domain for \(y=sin(x)\) until the segment is strictly increasing.
(b)
Using Figure 5.3.1 adjust the left/right ends of the domain for \(y=cos(x)\) until the segment is strictly decreasing.
(c)
Using Figure 5.3.1 adjust the left/right ends of the domain for \(y=tan(x)\) until the segment is strictly increasing.
Figure 5.3.1. Restricted Domains

Subsection 5.3.2 Differentiate Inverse Sine

\begin{align*} y = \amp \arcsin x. \\ x = \amp \sin y. \\ 1 = \amp (\cos y)y^\prime. \\ y^\prime = \amp \frac{1}{\cos y}. \\ y^\prime = \amp \frac{1}{\sqrt{1-x^2}}. \end{align*}
Figure 5.3.2. Triangle for Inverse Sine

Subsection 5.3.3 Algebra to the Rescue: Inverse Trig Anti-Derivatives

The following example begins with an integrand that looks similar to the derivative of inverse sine, but has coefficients other than one. Algebra enables us to transform it into the desired form.
\begin{align*} \int \frac{5}{\sqrt{16-32x^2}} dx = \amp \int \frac{5}{\sqrt{16(1-2x^2)}} dx \\ = \amp \int \frac{5}{4\sqrt{1-2x^2}} dx \\ = \amp \frac{5}{4} \int \frac{1}{\sqrt{1-2x^2}} dx \\ u^2 = \amp 2x^2 \\ u = \amp \sqrt{2}x \\ du = \amp \sqrt{2} dx \\ = \amp \frac{5}{4\sqrt{2}} \int \frac{\sqrt{2}}{\sqrt{1-2x^2}} dx \\ = \amp \frac{5}{4\sqrt{2}} \int \frac{1}{\sqrt{1-u^2}} du \\ = \amp \frac{5}{4\sqrt{2}} \arcsin u + C \\ = \amp \frac{5}{4\sqrt{2}} \arcsin(\sqrt{2}x) + C. \end{align*}

Subsection 5.3.4 Applications

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