Evaluate \[ \int\frac{dx}{\sqrt{x^2+25}} \qquad\text{and}\qquad \int\frac{dx}{(1+x^2)^{3/2}}. \]
Solution
For the first integral, the standard hyperbolic/trigonometric substitution gives \[ \boxed{\int\frac{dx}{\sqrt{x^2+25}} =\ln\!\left(x+\sqrt{x^2+25}\right)+C.} \] The logarithm’s argument is positive for every real \(x\).
For the second, \[ \frac{d}{dx}\left(\frac{x}{\sqrt{1+x^2}}\right) =\frac1{(1+x^2)^{3/2}}, \] so \[ \boxed{\int\frac{dx}{(1+x^2)^{3/2}} =\frac{x}{\sqrt{1+x^2}}+C.} \]