Prove the following reduction formula: integrate of (tan^(n)x) dx= (tan^(n-1)x)/(n-1) - integrate of (tan^(n-2))dx

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rcmath eNotes educator| Certified Educator

we need to show that `inttan^n(x)dx=(tan^(n-1)(x))/(n-1)-inttan^(n-2)(x)dx`

We know that `tan^2(x)=sec^2(x)-1`

hence we can rewrite our integral in the following manner:





In order to integrate the first part let u=tan(x) then du=sec^2(x)dx

Thus the above problem becomes








mlehuzzah eNotes educator| Certified Educator

`int "tan"^n x dx =`

`int ("tan" ^(n-2) x) ("tan" ^2 x) dx =`

`int ("tan" ^(n-2) x) ("sec"^2 x - 1)dx = `

`int ("tan" ^(n-2) x ) "sec"^2 x dx - int "tan" ^(n-2) x dx=`

To do the first integral, use a u substitution:

`u="tan" x` `du = "sec"^2 x dx`

`int ("tan" ^(n-2) x )"sec"^2 x dx = `

`int u^(n-2) du =`

`1/(n-1) u^(n-1) =`

`1/(n-1) "tan" ^(n-1) x`

Putting the pieces together, we get the reduction formula:

`int "tan"^n x dx = 1/(n-1) "tan" ^(n-1) x - int "tan" ^(n-2) x dx`


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