Question

Right, or wrong? Say which for each formula and give a brief reason for each answer. int(2x + 1)2 dx =(2x + 1)3/ 3 + C

Applications of integrals
ANSWERED
asked 2021-02-20
Right, or wrong? Say which for each formula and give a brief reason for each answer. \(\displaystyle\int{\left({2}{x}+{1}\right)}{2}{\left.{d}{x}\right.}={\left({2}{x}+{1}\right)}\frac{{3}}{{3}}+{C}\)

Answers (1)

2021-02-21
Step 1
Consider the integral, \(\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}\)
Step 2
To solve the given integrals,
\(\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}=\int{\left({4}{x}^{{2}}+{1}+{4}{x}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle={4}\int{x}^{{2}}{\left.{d}{x}\right.}+\int{1}\cdot{\left.{d}{x}\right.}+{4}\int{x}{\left.{d}{x}\right.}\)
\(\displaystyle={4}{\left(\frac{{{x}^{{3}}}}{{3}}\right)}+{x}+{4}{\left(\frac{{{x}^{{2}}}}{{2}}\right)}+{C}\)
\(\displaystyle=\frac{{{4}{x}^{{3}}}}{{3}}+{2}{x}^{{2}}+{x}+{C}\)
Step 3
\(\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}=\frac{{{\left({2}{x}+{1}\right)}^{{3}}}}{{{3}\cdot{2}}}+{C}\)
\(\displaystyle=\frac{{{8}{x}^{{3}}+{1}+{6}{x}{\left({2}{x}+{1}\right)}}}{{6}}+{C}\)
\(\displaystyle=\frac{{{8}{x}^{{3}}+{1}+{12}{x}^{{2}}+{6}{x}}}{{6}}+{C}\)
\(\displaystyle=\frac{{{8}{x}^{{3}}}}{{6}}+\frac{{{12}{x}^{{2}}}}{{6}}+{6}\frac{{x}}{{6}}+\frac{{1}}{{6}}+{C}\)
\(\displaystyle=\frac{{{4}{x}^{{3}}}}{{3}}+{2}{x}^{{2}}+{x}+\frac{{1}}{{6}}+{C}\)
Hence the given integral: \(\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}=\frac{{{\left({2}{x}+{1}\right)}^{{3}}}}{{3}}+{C}\) is wrong.
The right answer is: \(\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}=\frac{{{\left({2}{x}+{1}\right)}^{{3}}}}{{6}}+{C}\)
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