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

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 $$m_k^{\left({f+c}\right)} = \inf_{x \in \left[{x_{k - 1} \,. \, . \, x_k}\right]} \left({f \left({x}\right) + c}\right)$$, where $$k \in \left\{{0, 1, \ldots, n}\right\}$$.

So:

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It follows that:

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

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