Question

Which of the following integrals cannot be evaluated using a simple substitution? (4 points) Select one:

a. the integral of the square root of the quantity x minus 1, dx
b. the integral of the quotient of 1 and the square root of the quantity 1 minus x squared, dx
c. the integral of the quotient of 1 and the square root of the quantity 1 minus x squared, dx
d. the integral of x times the square root of the quantity x squared minus 1, dx

Answer 1

Answer:

B. and C.

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Integration

• Integrals
• Indefinite Integrals
• Integration Constant C

U-Substitution

Step-by-step explanation:

*Note:

It seems like B and C are both the same answer.

Let's define our answer choices:

a.  $\displaystyle \int {\sqrt{x - 1}} \, dx$

b.  $\displaystyle \int {\frac{1}{\sqrt{1 - x^2}}} \, dx$

c.  $\displaystyle \int {\frac{1}{\sqrt{1 - x^2}}} \, dx$

d.  $\displaystyle \int {x\sqrt{x^2 - 1}} \, dx$

Let's run u-substitution through each of the answer choices:

a.  $\displaystyle u = x - 1 \rightarrow du = dx \ \checkmark$

∴ answer choice A can be evaluated with a simple substitution.

b.  $\displaystyle u = 1 - x^2 \rightarrow du = -2x \ dx$

We can see that this integral cannot be evaluated with a simple substitution. In fact, this is a setup for an arctrig integral.

∴ answer choice B cannot be evaluated using a simple substitution.

C.  $\displaystyle u = 1 - x^2 \rightarrow du = -2x \ dx$

We can see that this integral cannot be evaluated with a simple substitution. In fact, this is a setup for an arctrig integral.

∴ answer choice C cannot be evaluated using a simple substitution.

D.  $\displaystyle u = x^2 - 1 \rightarrow du = 2x \ dx \ \checkmark$

Using a little rewriting and integration properties, this integral can be evaluated using a simple substitution.

∴ answer choice D can be evaluated using a simple substitution.

Out of all the choices, we see that B and C cannot be evaluated using a simple substitution.

∴ our answer choices should be B and C.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e