Evaluate the Integral (trig substitution)

[goosefraba]

I need help with the following problem:

integral of: du/u(sqrt(5-u^2))

Any help with this would be greatly appreciated. I am supposed to use trig substitution to solve it.

 

[galactus]

If you have to use trig sub,let $\displaystyle u=\sqrt{5}sin(t), \;\ du=\sqrt{5}cos(t)dt$

$\displaystyle \int\frac{1}{\sqrt{5}sin(t)\sqrt{5-(\sqrt{5}sin(t))^{2}}}\cdot \sqrt{5}cos(t)dt$

$\displaystyle \int\frac{\sqrt{5}cos(t)}{\sqrt{5}sin(t)\sqrt{5(1-sin^{2}(t))}}dt$

Now, finish simplifying it down. It whittle down nicely.

Remember, $\displaystyle 1-sin^{2}(t)=cos^{2}(t)$

If done correctly, you should get it down to $\displaystyle \frac{1}{\sqrt{5}}\int\frac{1}{sin(t)}dt=\frac{1}{\sqrt{5}}\int csc(t)dt$

Integrating csc is another matter, but it can be looked up or ran through a calculator.