How Do You Use Reduction Formulae for tan^n x?

[shamieh]

Can someone show me how I would "use" the reduction formulae for \(\displaystyle tan^n x\) ? I just want to see an example on when I would ever use it. A simple one will do.\(\displaystyle \frac{tan^{n-2}x}{n - 1} - \int tan^{n-2} x dx\)

 

[soroban]

Hello, shamieh!

Your formula is incorrect.

$$\int\tan^n\!x\,dx \;=\;\frac{\tan^{n-1}\!x}{n - 1} - \int\tan^{n-2}\!x\,dx$$Example: .$$\int\tan^4\!x\,dx$$

Substitute $$n=4$$ into the formula:

$$\int\tan^4\!x\,dx \;=\;\tfrac{1}{3}\tan^3\!x - \int \tan^2\!x\,dx$$

. . . . . . . . . $$=\;\tfrac{1}{3}\tan^3\!x - \int(\sec^2\!x -1)\,dx$$

. . . . . . . . . $$=\;\tfrac{1}{3}\tan^3\!x - \int\sec^2\!x\,dx + \int dx$$

. . . . . . . . . $$=\;\tfrac{1}{3}\tan^3\!x - \tan x + x + C$$

 

[shamieh]

Thank you sororaban! Also thank you for the PM the other day!