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. What is the total debt after 5 years?

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

Question 1

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

Accepted Answer

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

Question 2

When using substitution to evaluate $$\int 4x(x^2$+3)^5$$ \, dx$$, what is the best choice for u$?

Accepted Answer

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

Question 3

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

Accepted Answer

$$\frac{1}{2} e$$^{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. What is the total debt after 5 years?

Accepted Answer

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

Question 5

Which substitution would simplify the integral $$\int \frac{1}{x \ln x} \, dx$$?

Accepted Answer

$$u = \ln x$$

Question 6

Find $$\int \frac{2x}{(x^2$+1)^3$$} \, dx$ using substitution.

Accepted Answer

$$-\frac{1}{(x^2$+1)^2$$} + C$

Question 7

If $$u = x^2$ + 3x$$, what is $$\frac{du}{dx}$$?

Accepted Answer

$$2x + 3$$

Question 8

Which of the following integrals can be solved directly by substitution?

Accepted Answer

$$\int x \cos(x^2$) \, dx$$

Question 9

Evaluate $$\int \frac{1}{x^2$+1} \, dx.$$

Accepted Answer

$$\arctan(x) + C$$

Question 10

A company’s debt increases at a rate $$D'(t) = 1000 + 0.01t^2$. If the company starts with D(0) = 0$, what is the expression for total debt D(t$$)$$ after t$ years?

Accepted Answer

$$D(t) = 1000t + \frac{0.01}{3}t^3$

Question 11

Which of the following is NOT a valid step in the substitution method for integration?

Accepted Answer

Leave some $$x$$ terms in the integrand after substitution.

Question 12

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

Accepted Answer

$$\frac{1}{2} \sqrt{4x+1} + C$$

Question 13

Which substitution would you use to evaluate $$\int x^3$ \cos(x^4$) \, dx$$?

Accepted Answer

$$u = x^4$

Question 14

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

Accepted Answer

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

Question 15

If $$u = 4x + 1$$, what is $$dx in $$terms of $$du$$?

Accepted Answer

$$dx = \frac{du}{4}$$

Question 16

Evaluate $$\int \frac{1}{x^2$+3x+2} \, dx$$ using partial fractions.

Accepted Answer

$$\ln|x+2| - \ln|x+1| + C$$

Question 17

Which of the following is a real-world scenario where substitution is useful in business calculus?

Accepted Answer

Finding the total accumulated cost when the rate of spending is a nonlinear function of time.

Question 18

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

Accepted Answer

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

Question 19

Which of the following is the correct integral for $$\int (4x+1)^3$ \, dx$$ using substitution?

Accepted Answer

$$\frac{1}{4} \int u^3$ \, du$$

Integration by Substitution (Section 7.2) – Business Calculus Study Notes

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