# Evaluate the integral. \int_{-7}^{6}e^{-0.25x}dx

Evaluate the integral.
$$\displaystyle{\int_{{-{7}}}^{{{6}}}}{e}^{{-{0.25}{x}}}{\left.{d}{x}\right.}$$

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Coldst
Step 1
Evaluate the integral:
$$\displaystyle{\int_{{-{7}}}^{{{6}}}}{e}^{{-{0.25}{x}}}{\left.{d}{x}\right.}=-{\frac{{{1}}}{{{0.25}}}}{{\left[{e}^{{-{0.25}{x}}}\right]}_{{-{7}}}^{{{6}}}}$$
$$\displaystyle=-{\frac{{{1}}}{{{0.25}}}}{\left[{e}^{{-{0.25}\times{6}}}-{e}^{{-{0.25}\times{\left(-{7}\right)}}}\right]}$$
$$\displaystyle={4}{\left({e}^{{{1.75}}}-{e}^{{-{1.5}}}\right)}$$
$$\displaystyle\approx{22.126}$$
Step 2
Thus, the integral equals 22.126.
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Vaing1990
Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:
$$\displaystyle{\int_{{{a}}}^{{{b}}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({x}\right)}{{\mid}_{{{a}}}^{{{b}}}}={F}{\left({b}\right)}-{F}{\left({a}\right)}$$
Step 2: In this case, $$\displaystyle{f{{\left({x}\right)}}}={e}^{{-{0.25}{x}}}$$. Find its integral.
$$\displaystyle-{4}{e}^{{-{0.25}{x}}}{{\mid}_{{-{7}}}^{{{6}}}}$$
Step 3: Since $$\displaystyle{F}{\left({x}\right)}{{\mid}_{{{a}}}^{{{b}}}}={F}{\left({b}\right)}-{F}{\left({a}\right)}$$, expand the above into F(6)−F(−7):
$$\displaystyle-{4}{e}^{{-{0.25}\times{6}}}-{\left(-{4}{e}^{{-{0.25}\times-{7}}}\right)}$$
Step 4: Simplify.
$$\displaystyle-{4}{e}^{{-{1.5}}}+{4}{e}^{{{1.75}}}$$