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
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: \(\displaystyle\int{\left({2}{x}+{1}\right)}^{{2}}{\left.{d}{x}\right.}=\frac{{{\left({2}{x}+{1}\right)}^{{3}}}}{{6}}+{C}\)
0

Relevant Questions

asked 2021-03-07
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?
\(\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}\).
asked 2020-12-05
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
asked 2020-11-08
Evaluate the following integrals.
\(\displaystyle\int{\left({2}{x}^{{{3}}}-{x}^{{{2}}}+{3}{x}-{7}\right)}{\left.{d}{x}\right.}\)
asked 2021-03-09
Evaluate the following integral.
\(\displaystyle\int{2}{x}{\left({1}-{x}^{{-{3}}}\right)}{\left.{d}{x}\right.}\)
asked 2020-10-18
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?
asked 2021-01-28
Evaluate the integrals using a table of integrals.
\(\displaystyle\int{x}{{\sin}^{{-{{1}}}}{2}}{x}{\left.{d}{x}\right.}\)
asked 2021-02-25
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.
asked 2020-12-13
Find the following integral.
\(\displaystyle\int{\left({\frac{{{2}}}{{{x}^{{{3}}}}}}+{\frac{{{1}}}{{\sqrt{{{x}}}}}}\right)}{\left.{d}{x}\right.}\)
asked 2021-01-15
Evaluate the integral.
\(\displaystyle\int{\left(\frac{{2}}{{x}^{{3}}}+\frac{{1}}{\sqrt{{x}}}\right)}{\left.{d}{x}\right.}\)
asked 2021-02-26
given \(\displaystyle{y}=\frac{{1}}{{x}}\) is a solution \(\displaystyle{2}{x}^{{2}}{d}{2}\frac{{y}}{{\left.{d}{x}\right.}}+{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}-{3}{y}={0},{x}{>}{0}\)
a) Find a linearly independent solution by reduction the order approach
b) Show that 2 solutions are linearly independent
c) Write a general solution
...