In dealing with trigonometric substitution and integration

[Badgerspin]

Here's the equation:

∫(sqrt(2),2) (1/(x^3*sqrt(x^2 - 1))

I have the entire indefinite integral worked down to this (using x = a*secø):

ø/2 + 1/4 * sin2ø

Now I have the answer book, so I know that's right so far. What I don't understand is how it converted the points of the integral. Where we started with the integral from sqrt(2),2. The book is telling me that the points now being worked with are pi/4, pi/3.

How did we go from sqrt(2), 2 to pi/4, pi/3 ? I don't understand where that came from. Beyond that, how do I derive those points in general? I'm the first to admit that anything involving trig is by and far my weakest aspect of calculus.

Any help would be greatly appreciated.

 

[hunt_mat]

Try $$x=\sec\theta$$

 

[Badgerspin]

My problem is that I really don't know how they went from 2 to pi/3 and √2 to pi/3. I'm well aware of what substitution I needed to use. I have no idea what sec(ø) does to help answer this.

This is probably basic trig, but I'm finding that there are a lot of things in basic trig that I never learned that are now being applied in Calculus 2.

 

[Mark44]

Here's the equation:

∫(sqrt(2),2) (1/(x^3*sqrt(x^2 - 1))

Your notation is a bit on the inscrutable side, but this is what I think you're working with.
$$\int_{\sqrt{2}}^2 \frac{dx}{x^3 \sqrt{x^2 - 1}}$$

The limits of integration are sqrt(2) and 2.

I have the entire indefinite integral worked down to this (using x = a*secø):

ø/2 + 1/4 * sin2ø

Now I have the answer book, so I know that's right so far. What I don't understand is how it converted the points of the integral. Where we started with the integral from sqrt(2),2. The book is telling me that the points now being worked with are pi/4, pi/3.

How did we go from sqrt(2), 2 to pi/4, pi/3 ? I don't understand where that came from. Beyond that, how do I derive those points in general? I'm the first to admit that anything involving trig is by and far my weakest aspect of calculus.

Any help would be greatly appreciated.

If you make a substitution, including a trig substitution, in a definite integral, you can skip the step of undoing the substitution by changing the limits of integration.

With the substitution x = sec(theta), if x = sqrt(2), then theta = pi/4. If x = 2, then theta = pi/3. As a check, cos(pi/4) = sqrt(2)/2 = 1/sqrt(2) ==> sec(pi/4) = sqrt(2). Also, cos(pi/3) = 1/2 ==> sec(pi/3) = 2.