Reduction Formula for the Integral of ∫tan^n(x)dx

So far, we have worked through the steps to derive the reduction formulas (formulae) for the integrals of powers of sine and cosine. Let's now have a look at how we derive the reduction formula for integrating powers of the tangent function.

As it turns out, and counter intuitively, deriving this reduction formula is easier than for sine and cosine. The first step is the same - break up the integrand into a multiple of lower powers.

Let's reserve a tan-squared term and rewrite the integral as...

Now, with the Pythagorean Identity, the tan-squared term can be rewritten as...

...and thus, the integral becomes...

Now, for the first integral on the right-hand side, we can use a simple u-substitution if we let...

And thus, the first integral on the right-hand side becomes...

And thus far, we have...

To complete, all we need to do is back substitute tan(x) for u into the above expression.

And there you have it - we've derived the reduction formula for the integral of powers of tan(x).

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