Question

For \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx,\] which functions are chosen as the first function and the second function?

Options

• First function \[\sin^{-1}x\]; second function \[\frac{x}{\sqrt{1-x^2}}\]

• First function \[\frac{x}{\sqrt{1-x^2}}\]; second function \[\sin^{-1}x\]

• First function \[x\]; second function \[\frac{\sin^{-1}x}{\sqrt{1-x^2}}\]

• First function \[\sqrt{1-x^2}\]; second function \[x\sin^{-1}x\]

Answer 1

The first function is \(\sin^{-1}x\), an Inverse trigonometric function under LIATE. The second function is \(\frac{x}{\sqrt{1-x^2}}\), whose integral is found first by substitution.