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:

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


Proof

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_n}\right\}$ be a finite 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:

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

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


So:

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


It follows that:

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


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

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

$\blacksquare$


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