# Reduction formula, integration problem

## Homework Statement

Part of an example of using reduction formula, I won't post the whole question as I get most of it, just at the very end, magical things happen with the working and things disappear, as they usually do with integration:

The definite integral goes from 0 to pi/4

=>$$\int$$ tan$$^{n-2}$$ x sec$$^{2}$$ x dx
=>tan$$^{n-1} x/n -1$$

## The Attempt at a Solution

My question is, what happened to the sec$$^{2}$$ x in the integration process?
I see the tan got integrated, but I can't figure out how sec disappears

Related Calculus and Beyond Homework Help News on Phys.org
Hootenanny
Staff Emeritus
Gold Member
Notice that

$$\frac{d}{dx}\tan x = sec^2x$$

$$\Rightarrow dx = \frac{d\left(\tan x\right)}{\sec^2 x}$$

Ah...I see, so that's how the sec^2 x gets canceled out.

Thanks hootenanny!

Hootenanny
Staff Emeritus
Gold Member
Ah...I see, so that's how the sec^2 x gets canceled out.

Thanks hootenanny!
A pleasure.

Mark44
Mentor
Since the integral is a definite integral, you should end up with a number, or at least an expression that doesn't involve n. Your final expression can easily be evaluated at both limits of integration.

As a minor point, you're using an "implies" symbol (==>) incorrectly. The first integral doesn't "imply" the second; it's equal to it.

Mark44
Mentor
A pleasure.
It probably has been asked before, but what's the English equivalent of your signature line? I think I understand a few of the words.

disce quasi semper victurus vive quasi cras moriturus

speak? almost always of victory? live almost ?? (you?) die

Thanks

Hootenanny
Staff Emeritus