# Derivating an integral

I have a question about, how can i derivate an expression of the form $$F\left( u \right) = \int\nolimits_{C_1 }^{C_2 } {f\left( {ux} \right)\mathrm{d}x}$$ where $C_1 ,C_2$ are constants. I have no idea :/

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This is called differentiating under the integral sign. But since $C_1$ and $C_2$ are constants, the answer is simply $\int_{C_1}^{C_2} \frac{\partial}{\partial u}f(ux) dx$. – Srivatsan Sep 4 '11 at 14:14
..., which is $\int_{C_1}^{C_2}xf'(ux)\mathrm dx=\frac1u\left([xf(ux)]_{C_1}^{C_2}-\int_{C_1}^{C_2}f(ux)\mathrm dx\right)$ (where the last step is only valid for $u\neq0$). – joriki Sep 4 '11 at 14:30
But for example here \eqalign{ & f\left( a \right) = \int\limits_0^\infty {\frac{{Arc\tan \left( {ax} \right) - Arc\tan \left( x \right)}} {x}dx} \cr & \cr & \frac{d} {{da}}f\left( a \right) = \int\limits_0^\infty {\frac{{dx}} {{1 + a^2 x^2 }}} \cr} He forgets the 1/x in the expression he only considered Arctan(ax) but I think that he should derivate with respect to a $Arctan (ax) / x$ )= – August Sep 4 '11 at 14:32
@August, The new example is not of the form you gave in your question. And the differentiation seems ok. Note that you are (partial) differentiating w.r.t. $a$, not $x$. So the $x$ is a constant. Also differentiating $\arctan (ax)$ w.r.t. $a$ gives a spare $x$, thanks to chain rule. These two $x$'s cancel nicely. – Srivatsan Sep 4 '11 at 14:56

This operation is called differentiating under the integral sign. In the general case, $$F(u) = \int_{a(u)}^{b(u)} g(u,x) dx,$$ where the limits are functions of the parameter $u$, we have $$F'(u) = f(u, b(u)) b'(u) - f(u, a(u)) a'(u) + \int_{a(u)}^{b(u)} \frac{\partial}{\partial u} g(u,x) dx.$$ In your case, the limits are constants, and $g(u,x)$ has the special form $f(ux)$. So the first two terms drop out, giving: $$F'(u) = \int_{C_1}^{C_2} \frac{\partial}{\partial u} f(ux) dx = \int_{C_1}^{C_2} f'(ux) x dx,$$ thanks to the chain rule.
In general, one cannot "simplify" such answers any further, but in your case you can. Noting that $$F'(u) = \int_{C_1}^{C_2} \frac{x}{u} \frac{\partial}{\partial x} f(ux) dx,$$ and integrating by parts, we get: $$F'(u) = \left. \frac{x}{u} f(ux) \right|_{C_1}^{C_2} - \int_{C_1}^{C_2} \left(\frac{\partial}{\partial x} \frac{x}{u} \right) \cdot f(ux) dx = \ldots$$ I will leave it to you to complete the answer. You should be able to express the final integral in terms of $F(u)$ itself.
Edit: As @joriki notes in a comment, the "simplification" step is valid only if $u \neq 0$.