Question

Check whether the function f (x) = [x]+ e^x is integrable in [0,3].

Answer 1

Answer on Question #69113 – Math – Real Analysis

Question

Check whether the function $f(x) = [x] + e^x$ is integrable in $[0, 3]$.

Solution

The function $f_{1}(x) = e^{x}$ is obviously integrable in $[0, 3]$ because $e^{x}$ is continuous in $[0, 3]$ (see http://www.ams.jhu.edu/~prashant/continuous.pdf).

The function $f_{2}(x) = [x] = \begin{cases} 0, & 0 \leq x < 1 \\ 1, & 1 \leq x < 2 \\ 2, & 2 \leq x < 3 \\ 3, & x = 3 \end{cases}$ is obviously monotonic in $[0, 3]$. But monotonic function is Riemann integrable

(see https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf p. 13), so $f_{2}(x)$ is integrable.

We see that original function $f(x)$ admits decomposition $f(x) = f_{1}(x) + f_{2}(x)$ where $f_{1}$ and $f_{2}$ are both integrable in $[0, 3]$. Since the sum of two integrable functions is integrable

(see https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf p. 16-17) we conclude that the function $f(x) = [x] + e^{x}$ is integrable in $[0, 3]$.

**Answer**: The function $f(x) = [x] + e^x$ is integrable in $[0, 3]$.

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