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Let $I \subseteq \mathbb R$ be a limited interval, and define the measure of $I$ to be $$m(I) := 6\cdot\mathrm{length}(I\cap(3,5)) + 10\cdot\mathrm{count}(I\cap\{3,4,5\})$$ Where $\mathrm{count}$ is the standard counting measure. Evaluate the integral $$\int_{[1,4)} \frac{x}{x^2+1}dm(x)$$

I have managed to evaluate the measure of the domain of integration (it's $16$), but it defies me how I should perform the actual integration with respect to the given measure. After all, as far as I know, I can apply the FTC only with respect to the standard measure on $\mathbb R$—that is, $\mathrm{length}(\cdot)$ itself.

Any clarification would be welcome.

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    It’s just the sum of two integrals: the first is the ordinary one, but from $3$ to $4$ (and multiplied by $6$ of course) the second is $10$ times the function-value at $3$. No explanatory value here, I fear, but I hope from what I’ve said you can figure out why it’s what you want and why. – 2017-01-12

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Hint: Integration is linear in the integrand as well as the measure, in particular, for any measurable $f$ you have $$\int f(x)\, \textrm{d}(\mu+\nu)(x)=\int f(x)\, \textrm{d}\mu(x)+\int f(x)\, \textrm{d}\nu(x)$$ (as long as the right hand side is well-defined).