Contents of this Calculus 2 episode:

Trigonometric functions, Integration of trigonometric functions, Sine and cosine with exponential functions, Integration by parts step by step, Case of the snake biting its own tail.

If either α or β is odd, then we hit the jackpot.

Let's see a specific example.

We will take the factor with the odd exponent, and turn it into a product of one linear factor and some quadratic factors.

Then comes a minor trick.

Finally we multiply, and an old friend appears:

It doesn't pose a problem if the exponent of the cosx is of higher degree.

We will again take the factor with the odd exponent, and turn it into a product of one linear factor and some quadratic factors.

Then, we will need this cubic formula:

Finally comes this one, again:

If both α and β are even, then this method won't work.

In this case we will use the so called linearization formulae.

Let's see one like this, too.

If we recall, there was this thing:

That's what we will use now.

The case of the snake biting its own tail

In these cases we use partial integration, and then do some magic tricks.

Let's see this one, for instance:

Here, the casting doesn't matter, we can assign the roles either way, the result will be the same.

Integral by parts, again.

And by this, we returned to the original problem.

If we continued the integration now, then two consecutive partial integrations would take us back to the original problem.

And we could continue that till the end of time. But that would be quite boring, so instead, here is a trick.

The main idea of the trick is to write an equation of what we found so far.

And then solve the equation.

Let's see one more:

Calculus 2 episode