Definite Integral of Function plus Constant

Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.

Let $c$ be a constant.

Then:
 * $\ds \int_a^b \paren {\map f t + c} \rd t = \int_a^b \map f t \rd t + c \paren {b - a}$

Proof
Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.

Let $\map {L^{\paren {f + c} } } P$ be the lower sum of $\map f x + c$ on $\closedint a b$ belonging to $P$.

Let:
 * $\ds m_k^{\paren {f + c} } = \map {\inf_{x \mathop \in \closedint {x_{k - 1} } {x_k} } } {\map f x + c}$

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

So:

It follows that:

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