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

Caelan
2021-02-20
Answered

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

You can still ask an expert for help

hesgidiauE

Answered 2021-02-21
Author has **106** answers

Step 1

Consider the integral,$\int {(2x+1)}^{2}dx$

Step 2

To solve the given integrals,

$\int {(2x+1)}^{2}dx=\int (4{x}^{2}+1+4x)dx$

$=4\int {x}^{2}dx+\int 1\cdot dx+4\int xdx$

$=4\left(\frac{{x}^{3}}{3}\right)+x+4\left(\frac{{x}^{2}}{2}\right)+C$

$=\frac{4{x}^{3}}{3}+2{x}^{2}+x+C$

Step 3

$\int {(2x+1)}^{2}dx=\frac{{(2x+1)}^{3}}{3\cdot 2}+C$

$=\frac{8{x}^{3}+1+6x(2x+1)}{6}+C$

$=\frac{8{x}^{3}+1+12{x}^{2}+6x}{6}+C$

$=\frac{8{x}^{3}}{6}+\frac{12{x}^{2}}{6}+6\frac{x}{6}+\frac{1}{6}+C$

$=\frac{4{x}^{3}}{3}+2{x}^{2}+x+\frac{1}{6}+C$

Hence the given integral:$\int {(2x+1)}^{2}dx=\frac{{(2x+1)}^{3}}{3}+C$ is wrong.

The right answer is:$\int {(2x+1)}^{2}dx=\frac{{(2x+1)}^{3}}{6}+C$

Consider the integral,

Step 2

To solve the given integrals,

Step 3

Hence the given integral:

The right answer is:

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