Start with something simple like $f''(x) = -f(x) \implies f(x) = sin(x+c)$ where $c$ is some constant. Then move onto:
$$g(x) = \int_0^x{1 \over t^2 + 1}dt = [atan(x)]_0^x \equiv atan(x)$$
which is all very well, but what about doing a series expansion of $cos(x)$:
\[ cos(x) = \sum_{i=0}^\infty \frac{(ix)^{2i}}{(2i)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \epsilon_N(x) \]
for some recedingly small error $\epsilon_N(x)$.
More importantly, though, is what if it all goes horribly wrong? For example, I'm always getting my left and right mixed up, which can be fatal:
\[ \Xi(a,b) = \int \right\{ 1 + \sqrt{1 - { (x - a)^2 \over (x - c)^2 }} \left\} dx \]
which should have been:
\[ \Xi(a,b) = \int \left\{ 1 + \sqrt{1 - { (x - a)^2 \over (x - c)^2 }} \right\} dx \]
So much for $&--;$ and $-$, not to mention $\'$!