Q: A company issues debt at a rate of $$D'(t) = 1000 + 0.01t^2$ (dollars per year), where t$ is the number of years since the company was founded. How much debt has the company issued after 5 years?

$$5000 + 0.01 \times \frac{125}{3} = 5000 + 0.4167 \approx 5000.42$$

Q: A company’s cost function is $$C'(t) = 200 + 5t$$ dollars per year. What is the total cost incurred over the first 4 years?

$$200 \times 4 + 5 \times \frac{4^2$}{2} = 800 + 40 = 840$$

Question 1

Which of the following best describes the substitution method for integration?

Accepted Answer

It involves replacing a complicated expression in the integrand with a new variable to simplify the integral.

Question 2

Given the integral $$\int 4x(x^2$+3)^5$$ \, dx$, what is the best substitution to use?

Accepted Answer

$$u = x^2$ + 3$$

Question 3

Evaluate $$\int 2x \cos(x^2$) \, dx$$ using substitution.

Accepted Answer

$$\sin(x^2$) + C$$

Question 4

A company issues debt at a rate of $$D'(t) = 1000 + 0.01t^2$ (dollars per year), where t$ is the number of years since the company was founded. How much debt has the company issued after 5 years?

Accepted Answer

$$5000 + 0.01 	imes \frac{125}{3} = 5000 + 0.4167 \approx 5000.42$$

Question 5

Which of the following integrals can be solved using the substitution $$u = 3x + 2$$?

Accepted Answer

$$\int (3x+2)^4$ \, dx$$

Question 6

Find $$\int \frac{1}{x \ln x} \, dx$$ using substitution.

Accepted Answer

$$\ln|\ln x| + C$$

Question 7

If $$f(x) = \sin(x^2$)$$, what is $$\int 2x f(x) \, dx$$?

Accepted Answer

$$\sin(x^2$) + C$$

Question 8

Evaluate $$\int x e$$^{x^2}$$ \, dx$$ using substitution.

Accepted Answer

$$\frac{1}{2} e$$^{x^2}$$ + C$$

Question 9

Which of the following is the correct application of the Chain Rule for derivatives?

Accepted Answer

$$\frac{d}{dx} [f(g(x))] = f'(g(x)) g'(x)$$

Question 10

A business's revenue is given by $$R'(t) = 500e$$^{0.02t}$$ dollars per year, where t$ is the number of years since launch. What is the total revenue after 10 years?

Accepted Answer

$$500 \$$int_0$$^{10} e$$^{0.02t}$$ \, dt = 500 \left[ \frac{1}{0.02} e$$^{0.02t}$$ \right]_0$$^{10}$$ = 25000 (e$$^{0.2}$$ - 1)$$

Question 11

Which substitution would simplify the integral $$\int \frac{2x}{(x^2$+1)^3$$} \, dx$?

Accepted Answer

$$u = x^2$ + 1$$

Question 12

Evaluate $$\int \frac{1}{4x+1} \, dx$$ using substitution.

Accepted Answer

$$\frac{1}{4} \ln|4x+1| + C$$

Question 13

Which of the following is a necessary step when using substitution to solve an integral?

Accepted Answer

Express $$dx in $$terms of $$du.$$

Question 14

Find $$\int \frac{1}{x^2$+4x+5} \, dx$$ using substitution.

Accepted Answer

$$\arctan\left(\frac{x+2}{1}\right) + C$$

Question 15

A company’s cost function is $$C'(t) = 200 + 5t$$ dollars per year. What is the total cost incurred over the first 4 years?

Accepted Answer

$$200 	imes 4 + 5 	imes \frac{4^2$}{2} = 800 + 40 = 840$$

Question 16

Evaluate $$\int \sin(3x) \, dx$$ using substitution.

Accepted Answer

$$-\frac{1}{3} \cos(3x) + C$$

Question 17

Which of the following is the correct integral for $$\int e$$^{2x}$$ \, dx$$ using substitution?

Accepted Answer

$$\frac{1}{2} e$$^{2x}$$ + C$$

Question 18

If $$u = x^2$ + 3$$, what is $$du in $$terms of $$dx$$?

Accepted Answer

$$du = 2x \, dx$$

Question 19

Evaluate $$\int \frac{1}{\sqrt{1-x^2$}} \, dx$$ using substitution.

Accepted Answer

$$\arcsin(x) + C$$

Integration by Substitution (Section 7.2): Study Notes

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