# Alternative methods to solving the integral of [tan(x)]^6 * sec(x )

• August 21st 2010, 10:01 PM
ElBartoV
Alternative methods to solving the integral of [tan(x)]^6 * sec(x )
I was just curious to see if there was alternative methods to solving

$\displaystyle\int \tan^6(x)\sec(x)\,dx$

asides from using integrations by parts since it tends to yield a very lengthy solution. Although, I believe the reduction formula for sec(x) could probably be applied in this case instead of integration by parts

$\displaystyle\int\sec^n(x)\,dx = -\frac{1}{n-1}\cot^{n-1}(x) - \cot^{n-2}(x)$

but, still is there any other methods asides from these two that could be used?

(P.S. sorry if the format is not completely correct, this is the first time i have used LaTex)
• August 21st 2010, 10:07 PM
Chris L T521
Quote:

Originally Posted by ElBartoV
I was just curious to see if there was alternative methods to solving

$\displaystyle\int \tan^6(x)\sec(x)\,dx$

asides from using integrations by parts since it tends to yield a very lengthy solution. Although, I believe the reduction formula for sec(x) could probably be applied in this case instead of integration by parts

$\displaystyle\int\sec^n(x)\,dx = -\frac{1}{n-1}\cot^{n-1}(x) - \cot^{n-2}(x)$

but, still is there any other methods asides from these two that could be used?

(P.S. sorry if the format is not completely correct, this is the first time i have used LaTex)

I tried to fix the tex for you...did I interpret everything correctly? I don't think I did... XD
• August 21st 2010, 10:55 PM
Prove It
$\tan^2{x} = \sec^2{x} - 1$.

So $\tan^6{x} = (\tan^2{x})^3$

$= (\sec^2{x} - 1)^3$

$= \sec^6{x} - 3\sec^4{x} + 3\sec^2{x} - 1$.

Therefore $\int{\tan^6{x}\sec{x}\,dx} = \int{(\sec^6{x} - 3\sec^4{x} + 3\sec^2{x} - 1)\sec{x}\,dx}$

$= \int{\sec^7{x} - 3\sec^5{x} + 3\sec^3{x} - \sec{x}\,dx}$.

Now you can integrate term by term using the reduction method you posted above.
• August 22nd 2010, 06:00 AM
Jester
Quote:

Originally Posted by ElBartoV
I was just curious to see if there was alternative methods to solving

$\displaystyle\int \tan^6(x)\sec(x)\,dx$

asides from using integrations by parts since it tends to yield a very lengthy solution. Although, I believe the reduction formula for sec(x) could probably be applied in this case instead of integration by parts

$\displaystyle\int\sec^n(x)\,dx = -\frac{1}{n-1}\cot^{n-1}(x) - \cot^{n-2}(x)$

but, still is there any other methods asides from these two that could be used?

(P.S. sorry if the format is not completely correct, this is the first time i have used LaTex)

Although still long, you could write the integral as

$\displaystyle \int \frac{\sin^6x}{\cos^7x}dx = \int \frac{\sin^6x \cos x}{\cos^8x}dx$

then let $u = \sin x$ giving

$\displaystyle \int \frac{u^6}{(1-u^2)^4}du = \int \frac{u^6}{(1-u)^4(1+u)^4}du$

then partial fractions.
• August 22nd 2010, 12:05 PM
ElBartoV
The method of partial fractions looks to be a little less lengthy then using the reduction formula or at the very least it’s easier to keep track of all the terms with only the variable u (a whole bunch of factors of sec(x) can get confusing sometimes). Also thank you for the tex fix, I’m still getting use to the LaTex system.