Section 1.7 Integrals Resulting in Inverse Trigonometric Functions — Summary
References.
As with Section 1.6 , this section of the OpenStax text just introduces a few useful indefinite integrals, and then gives some example and practice with using them in combination with substitutions; often simple ones of the form \(u = ax\text{;}\) these notes just provide a brief guide to that.
The two very useful integrals here are
\begin{align*}
\int \frac{dx}{\sqrt{a^2 - x^2}} \amp = \arcsin(x/a) + C, \quad a > 0, \text{and}\\
\int \frac{dx}{a^2 + x^2} \amp = \frac{1}{a}\arctan(x/a) + C
\end{align*}
The third one
\begin{equation*}
\int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a}\sec^{-1}(|x/a|) + C, \quad a > 0
\end{equation*}
is also occasionally useful, but less often.
Study Guide.
Study Calculus Volume 1, Section 5.7 2 . All Examples and Checkpoint items are worth looking at; then do a few Exercises from each of the ranges 391–394, 397–400, and 411–414.
openstax.org/books/calculus-volume-1/pages/5-7-integrals-resulting-in-inverse-trigonometric-functionsopenstax.org/books/calculus-volume-1/pages/5-7-integrals-resulting-in-inverse-trigonometric-functions
