# Reduction Formulae Application

#### shamieh

##### Active member
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.

$$\displaystyle \frac{tan^{n-2}x}{n - 1} - \int tan^{n-2} x dx$$

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#### soroban

##### Well-known member
Hello, shamieh!

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

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

##### Active member
Thank you sororaban! Also thank you for the PM the other day!