Question

integrate 1/x^2√(a^2-x^2) dx

Answer 1

\[\int\frac{dx}{x^2\sqrt{a^2-x^2}}\]
\[x=a\sin u~~\Rightarrow~~dx=a\cos u~du\]
\[\int\frac{a\cos u}{a^2\sin^2u\sqrt{a^2-a^2\sin^2u}}~du\\
\int\frac{\cos u}{a\sin^2u\sqrt{a^2}\sqrt{1-\sin^2u}}~du\\
\frac{1}{a^2}\int\frac{\cos u}{\sin^2u\sqrt{\cos^2u}}~du\\
\frac{1}{a^2}\int\frac{du}{\sin^2u}\\
\frac{1}{a^2}\int\csc^2u~du\\
-\frac{1}{a^2}\cot u+C\\
-\frac{1}{a^2}\cot \left(\sin^{-1}\frac{x}{a}\right)+C\\
-\frac{\sqrt{a^2-x^2}}{a^2x}+C\]