True/Fälse: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $u=x^2+1$, then $du=2x$

(b) True or False: If $dv=x^{2}dx$, then $v=\frac{1}{3}{x}^3$

(c) True or False: We can apply integration by parts with $u=\ln x$and $dv=xdx$to the integral $\int \frac{\ln x}{x}dx$

(d) True or False: We can apply integration by parts with $u=x$and $dv=\ln xdx$to the integral $\int \frac{\ln x}{x}dx$

(e) True or False: Integration by parts has to do with reversing the product rule.

(f) True or False: Integration by parts is a good method for any integral that involves a product.

(g) True or False: In applying integration by parts, it is sometimes a good idea to choose $u$to be the entire integrand and let $dv=dx$

(h) True or False:${\int }_0^{3}x{e}^{x}dx=x{e}^x-{\int }_0^3{e}^{x}dx$

Given the expression$u=x^2+1$

We have

$$\begin{gathered} u=x^2+1 \\ du=2xdx \end{gathered}$$

So the statement is false

Given the expression$dv=x^{2}dx$

Integrating, we get

$$\begin{gathered} dv=x^{2}dx \\ v=\int x^{2}dx \\ v=\frac{x^3}{3} \end{gathered}$$

So the statement is true

Given$u=\ln x$

We have

$u=\ln x,du=\frac{1}{x}dx$

$$\begin{gathered} \int \frac{\ln x}{x}dx=\int udu \\ =\frac{u^2}{2} \\ =\frac{(\ln x{)}^2}{2} \\ =\frac{(\ln x{)}^2}{2} \end{gathered}$$

So the statement is false

Given the expression$\int \frac{\ln x}{x}dx$

We have

$u=\ln x,du=\frac{1}{x}dx$

$$\begin{gathered} \int \frac{\ln x}{x}dx=\int udu \\ =\frac{u^2}{2} \\ =\frac{{\left(\ln x\right)}^2}{2} \end{gathered}$$

So the statement is false

Given Integration by parts has to do with reversing the product rule.

The statement is false. Because the integration by parts is not always a good method for any integral that involves product rule. The substitution method also sometimes can be used to solve the integrals.

Given Integration by parts is a good method for any integral that involves a product

The statement is false. Because the integration by parts is not always a good method for any integral that involves product rule. The substitution method also sometimes can be used to solve the integrals.

Given In applying integration by parts, it is sometimes a good idea to choose u to be the entire integrand and let$dv=dx$

The statement is true

Given${\int }_0^{3}x{e}^{x}dx=x{e}^x-{\int }_0^3{e}^{x}dx$

The statement is false