Question 1

Evaluate the definite integral $$\int_{0}^{1} 10(2x-1)^4$ + 9(2x-1)^2$ - 8\ dx$$ using an appropriate substitution.

Accepted Answer

$$-3$$

Question 2

Which substitution is most appropriate for evaluating $$\int (3x^2$ + 4)(x^3$ + 4x - 8$$)^{14}$$\ dx$$?

Accepted Answer

$$u = x^3$ + 4x - 8$$

Question 3

After substituting $$u = 2x - 1 in $$the integral $$\int_{0}^{1} 10(2x-1)^4$ + 9(2x-1)^2$ - 8\ dx$$, what are the new bounds for $$u$$?

Accepted Answer

$$u = -1$$ \ to \ u = 1$

Question 4

Evaluate $$\int \frac{2x-1}{3x^2$ - 3x + 1}\ dx$$ using substitution.

Accepted Answer

$$\frac{1}{3} \ln|3x^2$ - 3x + 1| + C$$

Question 5

Given $$u = x - 2$$, rewrite $$x + 3 in $$terms of $$u$$ for the integral $$\int \frac{x+3}{(x-2$$)^{16}$$}\ dx.$$

Accepted Answer

$$u + 5$$

Question 6

Find $$\int \frac{x+3}{(x-2$$)^{16}$$}\ dx$$ using substitution and express your answer in terms of $$x.$$

Accepted Answer

$$-\frac{1}{14}(x-2$$)^{-14}$$ - \frac{1}{3}(x-2$$)^{-15}$$ + C$$

Question 7

Which of the following is the correct substitution for $$\int \frac{x^2$ + 1}{\sqrt{x-3}}\ dx$$?

Accepted Answer

$$u = x - 3$$

Question 8

After substituting $$u = x - 3 in$$ $$\int \frac{x^2$ + 1}{\sqrt{x-3}}\ dx$$, what is $$x^2 + 1$ in terms of u$?

Accepted Answer

$$(u + 3)^2$ + 1$$

Question 9

Evaluate $$\int \frac{x^2$ + 1}{\sqrt{x-3}}\ dx$$ using substitution.

Accepted Answer

$$\frac{2}{7}(x-3$$)^{7/2}$$ + \frac{12}{5}(x-3$$)^{5/2}$$ + \frac{20}{3}(x-3$$)^{3/2}$$ + C$$

Question 10

A chemical reaction's rate is modeled by $$R(x) = 3x^2$ + 4$$, and the total change in concentration is $$x^3 + 4x - 8$. What integral gives the total effect of the reaction, and what is its value?

Accepted Answer

$$\int (3x^2$ + 4)(x^3$ + 4x - 8$$)^{14}$$\ dx = \frac{1}{15}(x^3$ + 4x - 8$$)^{15}$$ + C$$

Question 11

Which of the following integrals requires the use of integration by substitution for efficient evaluation?

Accepted Answer

$$\int (3x^2$ + 4)(x^3$ + 4x - 8$$)^{14}$$\ dx$$

Question 12

Suppose you are modeling the cooling of a medical sample, and the temperature function is $$T(x) = 3x^2$ - 3x + 1. If $$the rate of change is proportional to $$2x-1$$, which integral would you use to find the total change in temperature?

Accepted Answer

$$\int \frac{2x-1}{3x^2$ - 3x + 1}\ dx$$

Question 13

Which of the following best describes the process of changing the bounds when using substitution in a definite integral?

Accepted Answer

Substitute the original $$x$$ bounds into the $$u$$ equation to find new $$u$$ bounds.

Question 14

If $$u = x - 3$$, what is $$x in $$terms of $$u$$?

Accepted Answer

$$x = u + 3$$

Question 15

Which of the following is the correct antiderivative for $$\int u$$^{14}$$\ du$$?

Accepted Answer

$$\frac{1}{15}u$$^{15}$$ + C$$

Question 16

A real-world application requires integrating $$\int \frac{2x-1}{3x^2$ - 3x + 1}\ dx. $$What is the most efficient method to solve this integral?

Accepted Answer

Use substitution with $$u = 3x^2$ - 3x + 1$$

Question 17

Which of the following is the correct result for $$\int u$$^{-15}$$\ du$$?

Accepted Answer

$$-\frac{1}{14}u$$^{-14}$$ + C$$

Question 18

Which of the following is the correct result for $$\int u$$^{-16}$$\ du$$?

Accepted Answer

$$-\frac{1}{15}u$$^{-15}$$ + C$$

Integration Techniques: Substitution and Powers

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