Integral of tan^3(x)

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...

(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...

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

And thus the integral becomes...

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

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...

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

And therefore...

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

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

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

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...

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...

Figure 1. tan³(x) and its primitive

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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/√e^2x+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

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