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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