Originally Posted by

**artvandalay11** I would like to start out by saying that I am completely against making u substutions for all integrals... Mr F asks with incredulity: ALL ......??!

I would much rather recognize a pattern, guess at an antiderivative, and then put a correction factor out in front (I know most universities don't teach it like this, but let's give it a shot)

$\displaystyle \int \sin{(5x)}dx$

We know the integral of $\displaystyle \sin x$ is $\displaystyle -\cos x$

So let's "guess" that the integral is $\displaystyle -\cos (5x)$

Now if we take the derivative of that, we get $\displaystyle 5\sin (5x)$, so we're off by a factor of 5, so let's put $\displaystyle \frac{1}{5}$ in front of our "guess" and we'll be good

So the answer is $\displaystyle -\frac{1}{5}\cos(5x)$