Integral of Function plus Constant

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Theorem

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

Let $c$ be a constant.

Then:

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

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:

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

It follows that:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle L^{\left({f+c}\right)} \left({P}\right)$	$=$	$\displaystyle \sum_{k=1}^n m_k^{\left({f+c}\right)} \left({x_k - x_{k - 1} }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sum_{k=1}^n m_k^{\left({f}\right)} \left({x_k - x_{k - 1} }\right) + c \sum_{k=1}^n \left({x_k - x_{k - 1} }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle L^{\left({f}\right)} \left({P}\right) + c \left({b - a}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	as $\displaystyle \sum_{k=1}^n \left({x_k - x_{k - 1} }\right)$ telescopes

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \int_a^b \left({f \left({t}\right) + c}\right)\ \mathrm dt$	$=$	$\displaystyle \sup_P \left({L^{\left({f+c}\right)} \left({P}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sup_P \left({L^{\left({f}\right)} \left({P}\right) + c \left({b - a}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sup_P \left({L^{\left({f}\right)} \left({P}\right)}\right) + c \left({b - a}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \int_a^b f \left({t}\right) \ \mathrm dt + c \left({b - a}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

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

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle L^{\left({f+c}\right)} \left({P}\right)\)	\(=\)	\(\displaystyle \sum_{k=1}^n m_k^{\left({f+c}\right)} \left({x_k - x_{k - 1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{k=1}^n m_k^{\left({f}\right)} \left({x_k - x_{k - 1} }\right) + c \sum_{k=1}^n \left({x_k - x_{k - 1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle L^{\left({f}\right)} \left({P}\right) + c \left({b - a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	as $\displaystyle \sum_{k=1}^n \left({x_k - x_{k - 1} }\right)$ telescopes

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \int_a^b \left({f \left({t}\right) + c}\right)\ \mathrm dt\)	\(=\)	\(\displaystyle \sup_P \left({L^{\left({f+c}\right)} \left({P}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sup_P \left({L^{\left({f}\right)} \left({P}\right) + c \left({b - a}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sup_P \left({L^{\left({f}\right)} \left({P}\right)}\right) + c \left({b - a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \int_a^b f \left({t}\right) \ \mathrm dt + c \left({b - a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)