Evaluate the integral. \int_{-2}^{2}(x+3)^{2}dx

impresijuzj 2021-11-22 Answered
Evaluate the integral.
\(\displaystyle{\int_{{-{2}}}^{{{2}}}}{\left({x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}\)

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Expert Answer

Robert Harris
Answered 2021-11-23 Author has 214 answers
Step 1
Given:
The integral
\(\displaystyle{\int_{{-{2}}}^{{{2}}}}{\left({x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}\)
Step 2
Use substitution to integrate
Let u=x+3
\(\displaystyle\Rightarrow{d}{u}={\left.{d}{x}\right.}\)
When x = – 2 , u = – 2 + 3 = 1
When x = 2 , u = 2 + 3 = 5
Substituting u=x+3, we get the integral
\(\displaystyle{\int_{{{1}}}^{{{5}}}}{u}^{{{2}}}{d}{u}\)
Step 3
To simplify further, use the power rule of integration
\(\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{\left({n}+{1}\right)}}}}+{C}\)
\(\displaystyle{\int_{{{1}}}^{{{5}}}}{u}^{{{2}}}{d}{u}={{\left[{\frac{{{u}^{{{3}}}}}{{{3}}}}\right]}_{{{1}}}^{{{5}}}}\)
\(\displaystyle={\left({\frac{{{5}^{{{3}}}}}{{{3}}}}\right)}-{\left({\frac{{{1}}}{{{3}}}}\right)}\)
\(\displaystyle={\frac{{{125}}}{{{3}}}}-{\frac{{{1}}}{{{3}}}}\)
\(\displaystyle={\frac{{{125}-{1}}}{{{3}}}}\)
\(\displaystyle={\frac{{{124}}}{{{3}}}}\)
Therefore,
\(\displaystyle{\int_{{-{2}}}^{{{2}}}}{\left({x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}={\frac{{{124}}}{{{3}}}}\)
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Susan Yang
Answered 2021-11-24 Author has 977 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, \(\displaystyle{f{{\left({x}\right)}}}={\left({x}+{3}\right)}^{{{2}}}\). Find its integral.
\(\displaystyle{\frac{{{x}^{{{3}}}}}{{{3}}}}+{3}{x}^{{{2}}}+{9}{x}{{\mid}_{{-{2}}}^{{{2}}}}\)
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(−2):
\(\displaystyle{\left({\frac{{{2}^{{{3}}}}}{{{3}}}}+{3}\times{2}^{{{2}}}+{9}\times{2}\right)}-{\left({\frac{{{\left(-{2}\right)}^{{{3}}}}}{{{3}}}}+{3}{\left(-{2}\right)}^{{{2}}}+{9}\times-{2}\right)}\)
Step 4: Simplify.
\(\displaystyle{\frac{{{124}}}{{{3}}}}\)
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