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

[shamieh]

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

 

[soroban]

Hello, shamieh!

Your formula is incorrect.

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