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

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:

inttan^n(x)dx=inttan^(n-2)(x)*tan^2(x)dx=

inttan^(n-2)(x)(sec^2(x)-1)dx=

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

inttan^(n-2)(x)sec^2(x)dx-inttan^(n-2)dx=

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

Thus the above problem becomes

intu^(n-2)du-inttan^(n-2)(x)dx=

[u^(n-1)]/(n-1)-inttan^(n-2)(x)dx=

[tan^(n-1)(x)]/(n-1)-inttan^(n-2)(x)dx

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### Videos

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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