Evaluate the definite integral. \int_{0}^{-2}(5x^{2}-4x+2)dx

Serotoninl7 2021-11-17 Answered
Evaluate the definite integral.
\(\displaystyle{\int_{{{0}}}^{{-{2}}}}{\left({5}{x}^{{{2}}}-{4}{x}+{2}\right)}{\left.{d}{x}\right.}\)

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Expert Answer

Anot1954
Answered 2021-11-18 Author has 7464 answers
Step 1
Here, we use the result of integration that the integral of \(\displaystyle{x}^{{{n}}}\ {i}{s}\ {x}^{{\frac{{{n}+{1}}}{{{n}+{1}}}}}\). and the solve the definite integral as shown on board.
\(\displaystyle{\int_{{{0}}}^{{-{2}}}}{\left({5}{x}^{{{2}}}-{4}{x}+{2}\right)}{\left.{d}{x}\right.}={{\left|{5}{\frac{{{x}^{{{3}}}}}{{{3}}}}-{2}{x}^{{{2}}}+{2}{x}\right|}_{{{0}}}^{{-{2}}}}\)
\(\displaystyle={\left({5}{\frac{{{\left(-{2}\right)}^{{{3}}}}}{{{3}}}}-{2}{\left(-{2}\right)}^{{{2}}}+{2}{\left(-{2}\right)}\right)}-{\left({5}{\frac{{{0}^{{{3}}}}}{{{3}}}}-{2}{\left({0}\right)}^{{{2}}}+{2}{\left({0}\right)}\right)}\)
\(\displaystyle=-{\frac{{{40}}}{{{3}}}}-{8}-{4}-{0}=-{\frac{{{76}}}{{{3}}}}\)
Step 2
Ans: -76/3
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Nancy Johnson
Answered 2021-11-19 Author has 7406 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, \(\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{2}}}-{4}{x}+{2}\right)}\). Find its integral.
\(\displaystyle{\frac{{{5}{x}^{{{3}}}}}{{{3}}}}-{2}{x}^{{{2}}}+{2}{x}{{\mid}_{{{0}}}^{{-{2}}}}\)
Step 3: Since \(\displaystyle{F}{\left({x}\right)}{{\mid}_{{{a}}}^{{{b}}}}={F}{\left({b}\right)}-{F}{\left({a}\right)}\), expand the above into F(−2)−F(0):
\(\displaystyle{\left({\frac{{{5}{\left(-{2}\right)}^{{{3}}}}}{{{3}}}}-{2}{\left(-{2}\right)}^{{{2}}}+{2}\times-{2}\right)}-{\left({\frac{{{5}\times{0}^{{{3}}}}}{{{3}}}}-{2}\times{0}^{{{2}}}+{2}\times{0}\right)}\)
Step 4: Simplify.
\(\displaystyle=-{\frac{{{76}}}{{{3}}}}\)
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