How to integrate ∫tan^2(x)sin(x)dx

This is an integral that would have had me quite frustrated in my high school days of mathematics because I would have seen that there were two parts to this integral, and without thinking I would have applied integration by parts.

And if you were to take this approach, you'll find that it leads to a dead end very quickly and you may even find there are some expressions that are not integrate-able using the usual methods. So I suggest you don't go down that road.

But what I'm trying to convey with this tutorial is that integration always requires a bit more thinking to achieve the result compared to differentiation. It is not simply a mechanical process.

And with integrating trigonometric formulas like this, it's always handy to know what the trigonometric identities are. So if we realize that...

\tan^2x=\sec^2x-1 = \frac{1}{\cos^2x} - 1

Now then, if we multiply this expression by sin(x), so we say...

\tan^2x\sin x = \frac{\sin x}{\cos^2x} - \sin x = \tan x \sec x - \sin x

Please watch the video for the full tutorial!