# 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

Right, or wrong? Say which for each formula and give a brief reason for each answer. $\int \left(2x+1\right)2dx=\left(2x+1\right)\frac{3}{3}+C$
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Step 1
Consider the integral, $\int {\left(2x+1\right)}^{2}dx$
Step 2
To solve the given integrals,
$\int {\left(2x+1\right)}^{2}dx=\int \left(4{x}^{2}+1+4x\right)dx$
$=4\int {x}^{2}dx+\int 1\cdot dx+4\int xdx$
$=4\left(\frac{{x}^{3}}{3}\right)+x+4\left(\frac{{x}^{2}}{2}\right)+C$
$=\frac{4{x}^{3}}{3}+2{x}^{2}+x+C$
Step 3
$\int {\left(2x+1\right)}^{2}dx=\frac{{\left(2x+1\right)}^{3}}{3\cdot 2}+C$
$=\frac{8{x}^{3}+1+6x\left(2x+1\right)}{6}+C$
$=\frac{8{x}^{3}+1+12{x}^{2}+6x}{6}+C$
$=\frac{8{x}^{3}}{6}+\frac{12{x}^{2}}{6}+6\frac{x}{6}+\frac{1}{6}+C$
$=\frac{4{x}^{3}}{3}+2{x}^{2}+x+\frac{1}{6}+C$
Hence the given integral: $\int {\left(2x+1\right)}^{2}dx=\frac{{\left(2x+1\right)}^{3}}{3}+C$ is wrong.
The right answer is: $\int {\left(2x+1\right)}^{2}dx=\frac{{\left(2x+1\right)}^{3}}{6}+C$