What are some examples where it is analytically easier to compute integrals than derivatives? I know that in general, there exist more functions which are integrable than there are functions which are differentiable (nowhere differentiable to be exact), at least in $C([0,1])$, by the Baire Category Theorem. However, in general most elementary functions we know tend to have a derivative, which is generally almost always easier to compute than the integral. I have read in many places that functions or simulations where we need to compute either the derivative or integral analytically, the integral is easier. I can't find or think of any examples. Does anyone have any to shed light into this claim? Thanks!
 A: I do not know, if this counts as an example, but let $f$ denote any elementary, but "complicated" function, for example 
$$ f(x) = \exp\bigl(\sin x + \sqrt[23]{\cos x + 34}\bigr) + \sqrt[x]{x + \sin x} $$
or whatever. Then define $g(x) := \frac{f'(\log x)}x$. Then $f$ is an elementary function, the antiderivative of $g$ is $f \circ \log$ (quite easy), and although it is possible to compute $g'$ analytically (first of course one has to compute $f'$), I will not do this here.
A: Here I offer a practical perspective as to why integration is much "easier" than differentiation. 
When performing simulations of real life problems we frequently have to replace exact calculations with numerical approximations. Even if an exact formula can be found, it is often unsuitable for practical calculations because of the limitations of finite precision arithmetic.
In general, numerical integration is much less vulnerable to round off errors than numerical differentiation. Two small examples will illustrate this. 


*

*(Differentiation) Define
\begin{equation}
(D_h f)(x) = \frac{f(x+h) - f(x-h)}{2h}.
\end{equation}
In exact arithmetic, we have 
\begin{equation}
f'(x) - (D_h f)(x) = - \frac{1}{6} f^{(3)}(\xi) h^2
\end{equation}
for some $\xi \in (x-h,x+h)$ (Taylor's formula and a mean value theorem). However, if the computed values of $f_0 = f(x-h)$ and $f_1= f(x+h)$ have been tarnished by previous round off errors, say,
\begin{equation}
\hat{f}_0 = f_0 (1 + \delta_0), \quad \hat{f}_1 = f_1 (1 + \delta_1)
\end{equation}
where $|\delta_i|$ are tiny, say, bounded by the unit round off error $u$ (a very optimistic assumption!), then we can not compute $D_h f$, but the best we can hope for is
\begin{equation}
\hat{D}_h f = \frac{ \hat{f}_1 -\hat{f}_0}{2h} = D_h + \frac{f_1 \delta_1 - f_0 \delta_0}{2h}
\end{equation}
for which we have the depressing estimate
\begin{equation}
\left| f'(x) - \hat{D}_h f(x) \right| \leq \frac{1}{6} \|f^{(3)}\|_\infty h^2 + \frac{\|f\|_\infty}{h} u.
\end{equation}
We see that it is totally unrealistic to expect $\hat{D}_h f$ to approximate $f'(x)$ with a relative error comparable with $u$. 

*(Integration) Consider the problem of computing
\begin{equation}
I = \int_a^b f(x) dx.
\end{equation}
The composite trapezoidal rule corresponding to the stepsize $h$ is given by
\begin{equation}
T_h f = \sum_{j=0}^{N-1} \frac{h}{2} \left( f(x_j) + f(x_{j+1}) \right)
\end{equation}
where $Nh = b - a$ and $x_j = a + jh$ for $j=0,1,2,\dotsc, N$. For the exact trapezoidal sum we have
\begin{equation}
I - T_h f = O(h^2)
\end{equation}
If the computed function values are tarnished by rounding error (no worse than before), then we can not compute $T_h$, but the best we can hope for is
\begin{equation}
\hat{T}_h f = \sum_{j=0}^{N-1} \frac{h}{2} \left [ f(x_j)(1 + \delta_j) + f(x_{j+1})(1 + \delta_{j+1}) \right] 
\end{equation}
However, the distance to our target $T_h$ is modest, as
\begin{equation}
\left| T_h - \hat{T}_h \right| \leq u \sum_{j=0}^{N-1} \frac{h}{2} \left( |f(x_j)| + |f(x_{j+1}| \right) \approx u \int_a^b |f(x)|dx
\end{equation}
It follows that if $f$ does not change sign, then we can compute $I$ with essentially the same relative error as we computed $f$.
It is in this very real and practical sense that integration is far easier to accomplish accurately than differentiation is. In both cases there are techniques for detecting, estimating and mitigating the effect of round off errors, but in the case of experimental data, numerical differentiation remains far more vulnerable to the effects of errors than numerical integration.
