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Two ways to solve this (sort of) power-reduction Trig integral

Integrate:

$$\int_{0}^{\frac{\pi}{4}}tan^{7}x\ sec^{2}x\ dx$$

Check out the solutions and discussion here.

L'Hôpital's Rule and Fundamental Theorem of Calculus (FTC) lurk!

A tricky trigonometric integral with a surprising solution