Question

evaluate integral using lul where appropriate

du/ (u sqrt(5-u^2))

Answer 1

I'd use trigonometric substitution here.

Answer 2

To be honest I can follow my professor in class, but doing it on my own is confusing me. What do u mean by substitution

Answer 3

it says to use ln u when appropriate. there aren't any substitutions

Answer 4

Let
\[u = \sqrt{5} \sin w\]
\[du = \sqrt{5} \cos w dw\]
\[\rightarrow \frac{1}{\sqrt{5}} \int\limits \frac{dw}{\sin w}\]
\[= \frac{1}{\sqrt{5}} \int\limits \csc w dw = -\frac{1}{\sqrt{5}} \ln |\csc w + \cot w| + C\]

Answer 5

\[\csc w + \cot w = \frac{1 + \cos w}{\sin w} = \frac{1 + \sqrt{1-\sin^2 w}}{\sin w}\]
\[= \frac{1 + \frac{\sqrt{5 - u^2}}{\sqrt{5}}}{\frac{u}{\sqrt{5}}} = \frac{\sqrt{5} + \sqrt{5-u^2}}{u}\]

Answer 6

final answer
\[\frac{1}{\sqrt{5}} \ln |u| - \frac{1}{\sqrt{5}} \ln |\sqrt{5} + \sqrt{5-u^2}| + C\]