Definite Integral of Function plus Constant

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a .. b}\right]$.

Let $c$ be a constant.

Then:
 * $\displaystyle \int_a^b \left({f \left({t}\right) + c}\right) dt = \int_a^b f \left({t}\right) dt + c \left({b - a}\right)$

Proof
Let $P = \left\{{x_0, x_1, x_2, \ldots, x_n}\right\}$ be a subdivision of $\left[{a .. b}\right]$.

Let $L^{\left({f+c}\right)} \left({P}\right)$ be the lower sum of $f \left({x}\right) + c$ on $\left[{a .. b}\right]$ belonging to $P$.

Let:
 * $\displaystyle m_k^{\left({f+c}\right)} = \inf_{x \in \left[{x_{k - 1} .. x_k}\right]} \left({f \left({x}\right) + c}\right)$

where $k \in \left\{{0, 1, \ldots, n}\right\}$.

So:

It follows that:

So from the definition of definite integral, it follows that: