Q. 1 - Calculus

Question

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 $d u=2 x$

(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 $d v=x d x$ 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 $d v=d x$

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

Step-by-Step Solution

Answer

Part (a) The statements is false,

Part (b)  true, 

Part (c) false, 

Part (d) false, 

Part (e) true, 

Part (f) false, 

Part (g) true, 

Part (h) false 

Part (a) Step 1: Given information

Given the expression $u=x^2+1$

Part (a) Step 2: Integrate the expression

We have

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

So the statement is false

Part (b) Step 1: Given information

Given the expression $dv=x^{2}dx$

Part (b) Step 2: Integrate the expression

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

Part (c) Step 1: Given information

Given $u=\ln x$

Part (c) Step 2: Integration by substitution

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

Part (d) Step 1: Given information

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

Part (d) Step 2: Integration by substitution

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

Part (e) Step 1: Given information

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

Part (e) Step 2: Checking the reason

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.

Part (f) Step 1: Given information

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

Part (f) Step 2: Checking the reason

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.

Part (g) Step 1: Given information

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

Part (g) Step 2: Checking the reason

The statement is true

Part (h) Step 1: Given information

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

Part (h) Step 2: Checking the reason

The statement is false