Feb 11, 2014 Thread starter #1 S shamieh Active member Sep 13, 2013 539 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. \(\displaystyle \frac{tan^{n-2}x}{n - 1} - \int tan^{n-2} x dx\) Last edited: Feb 11, 2014
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. \(\displaystyle \frac{tan^{n-2}x}{n - 1} - \int tan^{n-2} x dx\)
Feb 11, 2014 #2 S soroban Well-known member Feb 2, 2012 409 Hello, shamieh! Your formula is incorrect. [tex]\int\tan^n\!x\,dx \;=\;\frac{\tan^{n-1}\!x}{n - 1} - \int\tan^{n-2}\!x\,dx[/tex] Example: .[tex]\int\tan^4\!x\,dx[/tex] Substitute [tex]n=4[/tex] into the formula: [tex]\int\tan^4\!x\,dx \;=\;\tfrac{1}{3}\tan^3\!x - \int \tan^2\!x\,dx[/tex] . . . . . . . . . [tex]=\;\tfrac{1}{3}\tan^3\!x - \int(\sec^2\!x -1)\,dx [/tex] . . . . . . . . . [tex]=\;\tfrac{1}{3}\tan^3\!x - \int\sec^2\!x\,dx + \int dx[/tex] . . . . . . . . . [tex]=\;\tfrac{1}{3}\tan^3\!x - \tan x + x + C[/tex]
Hello, shamieh! Your formula is incorrect. [tex]\int\tan^n\!x\,dx \;=\;\frac{\tan^{n-1}\!x}{n - 1} - \int\tan^{n-2}\!x\,dx[/tex] Example: .[tex]\int\tan^4\!x\,dx[/tex] Substitute [tex]n=4[/tex] into the formula: [tex]\int\tan^4\!x\,dx \;=\;\tfrac{1}{3}\tan^3\!x - \int \tan^2\!x\,dx[/tex] . . . . . . . . . [tex]=\;\tfrac{1}{3}\tan^3\!x - \int(\sec^2\!x -1)\,dx [/tex] . . . . . . . . . [tex]=\;\tfrac{1}{3}\tan^3\!x - \int\sec^2\!x\,dx + \int dx[/tex] . . . . . . . . . [tex]=\;\tfrac{1}{3}\tan^3\!x - \tan x + x + C[/tex]
Feb 11, 2014 Thread starter #3 S shamieh Active member Sep 13, 2013 539 Thank you sororaban! Also thank you for the PM the other day!