Integral of tan^3(x)

in #steemstem • 7 years ago (edited)

A little while ago, I posted a solution to deriving the reduction formula for the integral of tann(x), that is, integer powers of the tangent function. We use the principles discussed in this post to find the integral of...

t1.png

(NB: below this only one of many ways to solving this problem)

The first step is to reserve a tan2(x) and express the integrand as a product of 2 lower powers of tan(x). Thus...

t2.png

Alright, the next trick up our sleeve is to realise that, by the Pythagorean Identity...

t3.png

And thus the integral becomes...

t4.png

So now we have 2 integrals. The second integral is one of the standard integrals...

t5.png

To understand how this integral comes about, you can view this video.

Now for the first integral, we can realise that sec2(x) is the derivative of tan(x), so we can let...

t6.png

And from our previous studies into differentials, the term du/dx is separable. Thus...

t7.png

And therefore...

t8.png

Ok, let's put everything back together...

t9.png

Finally, with u = tan(x), the integral of tan3(x) is...

t10.png

You may have also see the result written like this...

t11.png

Are these results equal to each other? Well, let's see. Again, by the Pythagorean identity, equation (1), the first result can be written as...

t12.png

Now, with C and D being arbitrary constants of integration, we can see that equations (1) and (2) are equivalent.

The original function and its primitive (its integral) is plotted in Figure 1 below...

t15.png
Figure 1. tan3(x) and its primitive

Calculus: Techniques of Integration

  1. Integral of ∫4/(x3 + 4x)dx - do we need partial fractions?
  2. Integral of ∫3x2+8/x3+4xdx with f'(x)/f(x)
  3. The integral of sec(x) - Part 1
  4. The integral of sec(x) - Part 2
  5. Reduction Formula for the Integral of ∫tann(x)dx
  6. Integral of tan3(x)
  7. Reduction formula for Integral of ∫sinn(x)dx
  8. Integral of 1/√e2x+1

Calculus: Techniques of Integration [Videos]

  1. [DTube] Integral of tan(x)
  2. [DTube] How do I integrate tan^2(x)?
  3. [DTube] Integration of ∫sec(x)dx

Please give me an Upvote and Resteem if you have found this tutorial helpful.

Please ask me a maths question by commenting below and I will try to help you in future posts.

Tip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3

Tip me at PayPal: https://paypal.me/MasterWu

#steemiteducation #mathematics #calculus #teamaustralia
7 years ago in #steemstem by
$3.02
  • Past Payouts $3.02
  • - Author $2.38
  • - Curators $0.64
24 votes
Reply 2
Sort:  
  • Trending
    • [-]
     7 years ago  

    I've always been fascinated by integration.
    You did good on this one.