Example 23.11.

1.

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

Let \(\, x = \sin \theta \,\) then \(\, dx = \cos \theta \,d\theta \)

\begin {align*} \int \frac {dx}{\sqrt {1 - x^2}} & = \int \frac {\cos \theta \,d \theta }{\sqrt {1 - \sin ^2\theta }}\\\\ & = \int \frac {\cos \theta }{\cos \theta }\,d\theta \\\\ & = \int d\theta \\\\ & = \theta + c \quad \text {but}\quad x = \sin \theta \,\implies \, \theta = \sin ^{-1}x\\\\ \therefore \quad \int \frac {dx}{\sqrt {1 - x^2}} & = \sin ^{-1} x + c\\ \end {align*}

2.

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

Let \(\, x = \tan \theta \,\) then \(\, dx = \sec ^2\theta \,d\theta \)

\begin {align*} \int \frac {dx}{1 + x^2} & = \int \frac {\sec ^2\theta \,d\theta }{1 + \tan ^2\theta }\\\\ & = \int \frac {\sec ^2\theta \,d\theta }{\sec ^2\theta }\\\\ & = \int d\theta \\\\ & = \theta + c\\ & = \arctan x + c\\\\ \end {align*}