Find the integral, full solution \int (2x+3)^{2}dx

Monique Slaughter 2022-01-04 Answered
Find the integral, full solution
\(\displaystyle\int{\left({2}{x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}\)

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Expert Answer

Mary Herrera
Answered 2022-01-05 Author has 4127 answers
Step 1
Given integral,
\(\displaystyle\int{\left({2}{x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}\)
Step 2
\(\displaystyle\int{\left({2}{x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}=\int{\left({4}{x}^{{{2}}}+{12}{x}+{9}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle=\int{4}{x}^{{{2}}}{\left.{d}{x}\right.}+\int{12}{x}{\left.{d}{x}\right.}+\int{9}{\left.{d}{x}\right.}\)
\(\displaystyle={4}\int{x}^{{{2}}}{\left.{d}{x}\right.}+{12}\int{x}{\left.{d}{x}\right.}+{9}\int{\left.{d}{x}\right.}\)
\(\displaystyle={4}{\frac{{{x}^{{{3}}}}}{{{3}}}}+{12}{\frac{{{x}^{{{2}}}}}{{{2}}}}+{9}{x}+{C}\) Since \(\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}}+{C},\ \ \ {n}\ne{1}\)
\(\displaystyle={\frac{{{4}{x}^{{{3}}}}}{{{3}}}}+{6}{x}^{{{2}}}+{9}{x}+{C}\)
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Raymond Foley
Answered 2022-01-06 Author has 1234 answers
Calculate:
\(\displaystyle\int{\left({2}{x}+{3}\right)}^{{{2}}}{\left.{d}{x}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}\int{u}^{{{2}}}{d}{u}\)
\(\displaystyle\int{u}^{{{2}}}{d}{u}\)
\(\displaystyle={\frac{{{u}^{{{3}}}}}{{{3}}}}\)
\(\displaystyle{\frac{{{1}}}{{{2}}}}\int{u}^{{{2}}}{d}{u}\)
\(\displaystyle={\frac{{{u}^{{{3}}}}}{{{6}}}}\)
\(\displaystyle={\frac{{{\left({2}{x}+{3}\right)}^{{{3}}}}}{{{6}}}}\)
\(\displaystyle={\frac{{{\left({2}{x}+{3}\right)}^{{{3}}}}}{{{6}}}}+{C}\)
Let's simplify:
\(\displaystyle{\frac{{{4}{x}^{{{3}}}}}{{{3}}}}+{6}{x}^{{{2}}}+{9}{x}+{C}\)
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karton
Answered 2022-01-11 Author has 8659 answers

\(\int(2x+3)^{2}dx \\\int 4x^{2}+12x+9dx \\\int 4x^{2}dx+\int 12xdx+\int 9dx \\\frac{4x^{3}}{3}+6x^{2}+9x \\\text{Add C} \\\text{Answer:} \\\frac{4x^{3}}{3}+6x^{2}+9x+C\)

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