Integral of tan^n (x), reduction fromula?

[David112234]

Homework Statement

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The reduction formula for tanⁿx is a confusing matter for me,
First how do you derive it? Let's use tan³ as an example
Here in an integral calculator when it gets to tan⁴(x) it uses the reduction formula

BUT

When I ask it to integrate Tan³(x) the answer resembles nothing like the reduction formula.

So why do you use the reduction formula for tan^4 but not tan^3?
how do you derive the reduction formula using integration by parts?

2. Homework Equations
integration by parts ∫udv=uv-∫vdu
reduction formula ?

The Attempt at a Solution

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∫tan^3 (x) dx

u= tan^3(x) dv=1
du= 3tan²(x)sec²(x)dx v = x

= xtan³ - ∫x3tan²(x)sec²(x)dx

again, ∫x3tan²(x)sec²(x)dx

u = x dv= tan³sec²
du = dx v =(tan ^4 )/¼

This is going no where to what the reduction formula is supposed to look like...

 

[vela]

If you apply the reduction formula to $\tan^3 x$, you get
$$\int \tan^3 x\,dx = \frac{\tan^2 x}{2} - \int \tan x\,dx.$$ You just have to recognize that since $\sec^2 x = \tan^2 + 1$, the two results only differ by a constant.

To derive the reduction formula, you don't want to use integration by parts. Instead, write $\tan^n x = \tan^{n-2} x \ \tan^2 x = (\tan^{n-2} x)(\sec^2 x-1)$ and go from there.