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

elchatosarapage 2021-11-23 Answered
Evaluate the integral.
\(\displaystyle{\int_{{-{7}}}^{{{6}}}}{e}^{{-{0.25}{x}}}{\left.{d}{x}\right.}\)

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Expert Answer

Coldst
Answered 2021-11-24 Author has 751 answers
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
Answered 2021-11-25 Author has 1594 answers
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.
\(\displaystyle-{4}{e}^{{-{1.5}}}+{4}{e}^{{{1.75}}}\)
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