Question 1

Which of the following best describes the purpose of integration by substitution in calculus?

Accepted Answer

To rewrite an integral in terms of a new variable so that the integrand matches a known formula for integration.

Question 2

Given $$y = f(x) = x^2$, what is the differential dy$?

Accepted Answer

$$dy = 2x dx$$

Question 3

Suppose you are asked to evaluate $$\int (2x + 2) e$$^{x^2 + 2x + 5}$$ dx. $$Which substitution would best simplify the integral?

Accepted Answer

$$u = x^2$ + 2x + 5$$

Question 4

If $$u = 4x^3$, what is du$?

Accepted Answer

$$du = 12x^2$ dx$$

Question 5

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

Accepted Answer

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

Question 6

A company finds that the marginal price $$p'(x) at a $$supply level of $$x$$ bottles is given by $$p'(x) = \frac{1}{3x + 25}. If $$the distributor supplies 75 bottles at $$p(75) = 1.60$$, what is the price-supply equation $$p(x)$$?

Accepted Answer

$$p(x) = \ln|3x + 25| + C$$

Question 7

Which step is NOT part of the integration by substitution procedure?

Accepted Answer

Express the integrand entirely in terms of the original variable.

Question 8

Given $$w = h(t) = \ln(4 + 5t)$$, what is the differential $$dw$$?

Accepted Answer

$$dw = \frac{5}{4 + 5t} dt$$

Question 9

Evaluate $$\int \frac{1}{x + 6} dx$$ using substitution.

Accepted Answer

$$\ln|x + 6| + C$$

Question 10

If $$u = 5 - 2x^2$, what is du$?

Accepted Answer

$$du = -4x dx$$

Question 11

Evaluate $$\int \frac{2x}{x^2$ + 5} dx$$ using substitution.

Accepted Answer

$$\ln|x^2$ + 5| + C$$

Question 12

Which of the following is the correct substitution for evaluating $$\int x^2$ \cos(x^3$ - 5) dx$$?

Accepted Answer

$$u = x^3$ - 5$$

Question 13

After making the substitution $$u = x^3$ - 5$$, what adjustment must be made to the integral $$\int x^2$ \cos(x^3$ - 5) dx$$?

Accepted Answer

Multiply the integrand by $$\frac{1}{3} to $$account for $$du = 3x^2$ dx$$

Question 14

Evaluate $$\int \frac{1}{3x + 25} dx$$ using substitution.

Accepted Answer

$$\frac{1}{3} \ln|3x + 25| + C$$

Question 15

Which of the following is a valid general indefinite integral formula for substitution?

Accepted Answer

$$\int f'(g(x)) g'(x) dx = f(g(x)) + C$$

Question 16

A medical researcher models the rate of drug absorption as $$A'(t) = 4t e$$^{2t^2}$$. What is the total absorption A(t$$)$$?

Accepted Answer

$$2 e$$^{2t^2}$$ + C$$

Question 17

If $$u = x + 6$$, what is $$x in $$terms of $$u$$?

Accepted Answer

$$x = u - 6$$

Question 18

Which of the following is the correct differential for $$u = e$$^{3x}$$?

Accepted Answer

$$du = 3 e$$^{3x}$$ dx$$

Question 19

Evaluate $$\int \frac{1}{4 + 5t} dt$$ using substitution.

Accepted Answer

$$\frac{1}{5} \ln|4 + 5t| + C$$

Integration by Substitution: Business Calculus Study Notes

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