For \(x>1\), evaluate \[ \int x\,\sec^{-1}(x)\,dx \qquad\text{and}\qquad \int x\,\csc^{-1}(x)\,dx. \]
Solution
For \(x>1\), \[ \frac{d}{dx}\sec^{-1}x=\frac1{x\sqrt{x^2-1}}, \qquad \frac{d}{dx}\csc^{-1}x=-\frac1{x\sqrt{x^2-1}}. \] Integration by parts gives \[ \boxed{\int x\sec^{-1}x\,dx =\frac12\left(x^2\sec^{-1}x-\sqrt{x^2-1}\right)+C,} \] and \[ \boxed{\int x\csc^{-1}x\,dx =\frac12\left(x^2\csc^{-1}x+\sqrt{x^2-1}\right)+C.} \]