Chapter 7, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. The integral ${\displaystyle \int x^2{e}^{2x}dx}$ can be evaluated using integration by parts.
2. b. To evaluate the integral ${\displaystyle \int \frac{dx}{\sqrt{x^2-100}}}$ analytically, it is best to use partial fractions.
3. c. One computer algebra system produces ${\displaystyle \int 2\sin x\cos xdx={\sin }^{2}x}$. Another computer algebra system produces ${\displaystyle \int 2\sin x\cos xdx=-{\cos }^{2}x}$. One computer algebra system is wrong (apart from a missing constant of integration).
4. d. ${\displaystyle \int 2\sin x\cos xdx=-\frac{1}{2}\cos 2x+C.}$
5. e. The best approach to evaluating ${\displaystyle \int \frac{x^3+1}{3x^2}dx}$ is to use the change of variables u = x³ + 1.

To determine

Whether the statement, “The integral ${\displaystyle \int x^2}{e}^{x}dx$ can be evaluated using integration by parts”, is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

Integration by parts:

For the differentiable functions u and v, the integration by parts is given as follows.

$\begin{array}{l}{\displaystyle \int u\left(x\right)v^{\prime }}\left(x\right)dx=u\left(x\right)v\left(x\right)-{\displaystyle \int v\left(x\right)}{u}^{\prime }\left(x\right)dx\\ {\displaystyle \int udv=uv-{\displaystyle \int vdu}}\end{array}$

Calculation:

The given integral is ${\displaystyle \int x^2}{e}^{x}dx$.

Now, check whether the given integral can be solved by integration by parts as shown below.

$\begin{array}{c}{\displaystyle \int x^2}{e}^{x}dx=x^2\cdot e^x-{\displaystyle \int 2x\cdot e^{x}dx}\\ =x^2\cdot e^x-2{\displaystyle \int x}\cdot e^{x}dx\\ =x^2\cdot e^x-2\left[x\cdot e^x-{\displaystyle \int e^x}\right]\\ =x^2\cdot e^x-2x\cdot e^x+e^x+C\\ =e^x\left(x^2-2x+1\right)+C\end{array}$

Therefore, observe that the given integral can be solved by the method of integration by parts.

Thus, the statement is true.

To determine

Whether the statement, “To evaluate the integral ${\displaystyle \int \frac{dx}{\sqrt{x^2-100}}}$ analytically, it is best to use partial fractions”, is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

By a trigonometric substitution, ${\displaystyle \int \frac{dx}{\sqrt{x^2-a^2}}}={\cos }^{-1}\left(\frac{x}{a}\right)+C$.

Apply the above formula in to the given integral as, ${\displaystyle \int \frac{dx}{\sqrt{x^2-100}}}={\cos }^{-1}\left(\frac{x}{10}\right)+C$.

Therefore, it can be observed that direct trigonometric substitution is more easy than to apply the partial fractions for solving the given integral

Thus, the statement is false.

To determine

Whether the statement, “One computer algebra system produces ${\displaystyle \int 2\sin x\cos xdx={\sin }^{2}x}$ and another computer algebra system produces ${\displaystyle \int 2\sin x\cos xdx=-{\cos }^{2}x}$ such that one computer system is wrong”, is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The two computer algebra system shows the value of integral is as shown below.

$\begin{array}{l}{\displaystyle \int 2\sin x\cos xdx={\sin }^{2}x}\\ {\displaystyle \int 2\sin x\cos xdx=-{\cos }^{2}x}\end{array}$

Calculate the first integral as follows.

$\begin{array}{c}{\displaystyle \int 2\sin x\cos xdx}=2{\displaystyle \int \sin x\cos xdx}\left(\text{Put sin}x=u,\text{ cos}xdx=du\right)\\ =2{\displaystyle \int udu}\\ =2\frac{u^2}{2}+C\\ =\left({\sin }^{2}x\right)+C\end{array}$

Also, it is known that ${\sin }^{2}x=1-{\cos }^{2}x$.

Substitute the above known value in the above integration and solve as follows.

$\begin{array}{c}{\displaystyle \int 2\sin x\cos xdx}={\sin }^{2}x+C\\ =1-{\cos }^{2}x+C\end{array}$

Note that both values are the same.

Thus, the given statement is false.

To determine

Whether the statement “${\displaystyle \int 2\sin x\cos xdx=-\frac{1}{2}}\cos 2x+C$”, is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The given equation is ${\displaystyle \int 2\sin x\cos xdx=-\frac{1}{2}}\cos 2x+C$.

Integrate the given function as follows.

$\begin{array}{c}LHS={\displaystyle \int 2\sin x\cos xdx}\\ ={\displaystyle \int \sin 2xdx}\\ =-\frac{1}{2}\cos 2x+C\\ =RHS\end{array}$

Thus, the statement is true.

To determine

Whether the statement, “The best approach to evaluating ${\displaystyle \int \frac{x^3+1}{3x^2}}dx$ is to use the change of variables $u=x^3+1$”, is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Integrate the given integral as follows.

$\begin{array}{c}{\displaystyle \int \frac{x^3+1}{3x^2}}dx={\displaystyle \int \frac{x^3}{3x^2}}dx+{\displaystyle \int \frac{1}{3x^2}}dx\\ =\frac{1}{3}\left[{\displaystyle \int xdx+{\displaystyle \int x^{-2}dx}}\right]+C\\ =\frac{1}{3}\left[\frac{x^2}{2}-\frac{1}{x}\right]+C\end{array}$

Now, calculate the value of the integral using substitution as shown below.

${\displaystyle \int \frac{x^3+1}{3x^2}}dx={\displaystyle \int \frac{u}{3x^2}dx\left(\text{Put }u=x^3+1,3x^{2}dx=du\right)}$

From the above equation, note that the substitution $u=x^3+1$ is not possible to solve the equation.

Thus, the statement is false.