DITF - Differentiation and Integration of Inverse Trigonometric Functions Lesson

Differentiation and Integration of Inverse Trigonometric Functions

Given certain restrictions on the domain and range text annotation indicator , each of the six trigonometric functions is one-to-one and has an inverse, which are collectively referred to as inverse trigonometric functions. These inverse pairs are transcendental functions. Graphs of trigonometric functions and their inverses may be reviewed in the Preparation for Calculus module. Motivation for inverse trigonometric functions is introduced in the presentation below.

Evaluating Inverse Trigonometric Functions

A review of how to evaluate the most commonly used inverse trigonometric functions, arcsin, arctan, and arccos, follows.

The remaining three inverse trigonometric functions, arcsec, arccot, and arccsc, are not used as frequently since it is possible to express them in terms of their corresponding reciprocal relationships.

Derivatives of Inverse Trigonometric Functions

Although inverse trigonometric functions are transcendental, their derivatives are algebraic, as was the derivative of the natural logarithmic function. Derivations of derivates of four inverse trigonometric functions, examples of how to apply them, and a rules summary of derivatives of inverse trigonometric functions follow. Observe that the derivatives of the cofunctions are the negative of each other.

View a presentation that supplies a proof of the derivative of $LaTeX: \sin^{-1}x$ $\sin^{-1}x$ and examples related to other inverse trigonometric functions.

Explore finding derivatives of a variety of inverse trigonometric functions by CLICKING HERE Links to an external site..

Integrals of Inverse Trigonometric Functions

Each of the derivative formulas for inverse trigonometric functions has a related integral. Because the derivative formulas occur in opposite pairs of cofunctions, integrals for only three of the inverse trigonometric functions are typically used. Below summarizes the commonly used integrals of inverse trigonometric functions and provides an example of each.

*The last formula using inverse sec is wrong. It should have a 1/a in front*

Example 1

Determine the integral: text annotation indicator $LaTeX: \int\frac{dx}{\sqrt[]{4-x^2}}$ $\int\frac{dx}{\sqrt[]{4-x^2}}$

Solution: $LaTeX: \int\frac{dx}{\sqrt[]{4-x^2}}=arc\sin\frac{x}{2}+C\\ a^2=4\\ a=2\\ u=x\\ du=dx$ $\int\frac{dx}{\sqrt[]{4-x^2}}=arc\sin\frac{x}{2}+C\\ a^2=4\\ a=2\\ u=x\\ du=dx$

Example 2

Determine the integral: text annotation indicator $LaTeX: \int\frac{1}{1+9x^2}dx$ $\int\frac{1}{1+9x^2}dx$

Solution: $LaTeX: \int\frac{1}{1+9x^2}dx=\frac{1}{3}\int\frac{1}{1+u^2}du=\frac{1}{3}\arctan u+C=\frac{1}{3}\arctan3x+C\\ u=3x\\ du=3dx\\ \frac{du}{3}=dx$ $\int\frac{1}{1+9x^2}dx=\frac{1}{3}\int\frac{1}{1+u^2}du=\frac{1}{3}\arctan u+C=\frac{1}{3}\arctan3x+C\\ u=3x\\ du=3dx\\ \frac{du}{3}=dx$

Example 3

Determine the integral: text annotation indicator $LaTeX: \int\frac{dx}{x\ln x\sqrt[]{\left(\ln x\right)^2-1}}$ $\int\frac{dx}{x\ln x\sqrt[]{\left(\ln x\right)^2-1}}$

Solution: $LaTeX: \int\frac{dx}{x\:\ln x\sqrt[]{\left(\ln x\right)^2-1}}=\sec^{-1}\left|\ln x\right|+C\\ u^2=\left(\ln x\right)^2\\ u=\ln x\\ du=\frac{dx}{x}$ $\int\frac{dx}{x\:\ln x\sqrt[]{\left(\ln x\right)^2-1}}=\sec^{-1}\left|\ln x\right|+C\\ u^2=\left(\ln x\right)^2\\ u=\ln x\\ du=\frac{dx}{x}$

View the presentation involving the basics of integration of inverse trigonometric functions.

It is rare when problems involving inverse trigonometric functions do not require a substitution of variables (u-substitution). The next presentation focuses on using substitution to integrate several inverse trigonometric functions.

One of the motivations for evaluating definite integrals of inverse trigonometric functions is to determine the area under the graph of an inverse trigonometric function. The next presentation includes evaluation of definite integrals and the technique of completing the square, when integrands contain quadratic functions.

Differentiation and Integration of Inverse Trigonometric Functions Practice

Differentiation and Integration of Inverse Trigonometric Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of differentiation and integration of inverse trigonometric functions.

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