Can I separate a constant within an integral?

Question

Question

I was wondering if it was possible to remove a constant out of an integral?

example integral: $\int_1^7(f(x)+5)=\int_1^7(f(x))+\int_1^7(5)$

So my question is whether the above statement is possible?

Answer 1

Yes, it is valid and is the sum rule for integrals. This is due to the linearity of the integral. Rather than thinking in terms of proving this as true analytically, think about this graphically. Given $f(x)=x^2$, $\;f(x)+5=x^2+5$:

If we want to find:

$$\int_0^2 f(x)+5 \,\mathrm dx$$

By the sum rule, we can split it up:

$$\int_0^2 f(x) \,\mathrm dx + \int_0^2 5 \,\mathrm dx$$

And each integral represents a separate area:

Once we split the integral up, the first integral gives us the blue part, while the latter one the green part. This makes sense because adding a constant $5$ shifts the graph up $5$ units. Once we integrate, the whole area under the graph will be the original area under the graph, plus a new rectangle of height $5$, because the area under a constant function is a rectangle.

This then can be extended for arbitrary integrands because the area under a curve can be split up in the same way as above into two integrals, one for each integrand.