# 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

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

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: $$\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}=\frac{{{\left({2}{x}+{1}\right)}^{{3}}}}{{6}}+{C}$$

### Relevant Questions

Consider the integral as attached, To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals for the given integral to converge?
$$\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}$$.
Which of the following integrals are improper integrals?
1.$$\displaystyle{\int_{{{0}}}^{{{3}}}}{\left({3}-{x}\right)}^{{{2}}}{\left\lbrace{3}\right\rbrace}{\left.{d}{x}\right.}$$
2.$$\displaystyle{\int_{{{1}}}^{{{16}}}}{\frac{{{e}^{{\sqrt{{{x}}}}}}}{{\sqrt{{{x}}}}}}{\left.{d}{x}\right.}$$
3.$$\displaystyle{\int_{{{1}}}^{{\propto}}}{\frac{{{3}}}{{\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}}}}{\left.{d}{x}\right.}$$
4.$$\displaystyle{\int_{{-{2}}}^{{{2}}}}{3}{\left({x}+{1}\right)}^{{-{1}}}{\left.{d}{x}\right.}$$
a) 1 only
b)1 and 2
c)3 only
d)2 and 3
e)1,3 and 4
f)All of the integrals are improper
Evaluate the following integrals.
$$\displaystyle\int{\left({2}{x}^{{{3}}}-{x}^{{{2}}}+{3}{x}-{7}\right)}{\left.{d}{x}\right.}$$
Evaluate the following integral.
$$\displaystyle\int{2}{x}{\left({1}-{x}^{{-{3}}}\right)}{\left.{d}{x}\right.}$$
Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must $$\displaystyle\int{0}^{{2}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$$ lie? Which property of integrals allows you to make your conclusion?
Evaluate the integrals using a table of integrals.
$$\displaystyle\int{x}{{\sin}^{{-{{1}}}}{2}}{x}{\left.{d}{x}\right.}$$
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task. $$\displaystyle{\left({1},{3}\right)},{\left({2},{6}\right)},{\left({3},{12}\right)},{\left({4},{24}\right)}$$
Part A: Is this data modeling an algebraic sequence or a geometric sequence? Explain your answer.
Part B: Use a recursive formula to determine the time she will complete station 5.
Part C: Use an explicit formula to find the time she will complete the 9th station.
$$\displaystyle\int{\left({\frac{{{2}}}{{{x}^{{{3}}}}}}+{\frac{{{1}}}{{\sqrt{{{x}}}}}}\right)}{\left.{d}{x}\right.}$$
$$\displaystyle\int{\left(\frac{{2}}{{x}^{{3}}}+\frac{{1}}{\sqrt{{x}}}\right)}{\left.{d}{x}\right.}$$
given $$\displaystyle{y}=\frac{{1}}{{x}}$$ is a solution $$\displaystyle{2}{x}^{{2}}{d}{2}\frac{{y}}{{\left.{d}{x}\right.}}+{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}-{3}{y}={0},{x}{>}{0}$$