Question 1

Which of the following integrals is most appropriate for direct $$u-$$substitution?

Accepted Answer

$$\int \sqrt{x+3} dx$$

Question 2

What is the first step in the $$u-$$substitution process for evaluating $$\int 2x \sqrt{x^2$+1} dx$$?

Accepted Answer

Set $$u = x^2$ + 1$$

Question 3

Evaluate $$\$$int_0$$^{\pi/3} \sin(2	heta) d	heta$$ using $$u-$$substitution.

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 4

Which substitution would simplify the integral $$\int \sec(3x) 	an(3x) dx$$?

Accepted Answer

$$u = 3x$$

Question 5

A pharmaceutical company needs to calculate the area under the curve $$y = x^2$ (1-x$$)^{1/2}$$ from x=0$ to x=1$ to determine the total drug released over time. Which u$-substitution should be used?

Accepted Answer

$$u = 1-x$$

Question 6

Evaluate $$\int y (y-1$$)^{17}$$ dy$$ using $$u-$$substitution.

Accepted Answer

$$\frac{1}{18} y (y-1$$)^{18}$$ - \frac{1}{324} (y-1$$)^{18}$$

Question 7

A medical researcher models the concentration of a drug in the bloodstream as $$C(x) = \frac{x^2$}{(x^3$+2)^3$$}$$. To find the total concentration from x=-1$ to x=2$, which substitution is most effective?

Accepted Answer

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

Question 8

Which of the following is the correct evaluation of $$\int \sec^2$(3x) dx$$?

Accepted Answer

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

Question 9

A civil engineer needs to compute $$\$$int_2$$^6 \frac{x}{\sqrt{2x-3}} dx to $$determine the stress distribution along a beam. What is the value of this definite integral?

Accepted Answer

$$\frac{2}{3}[(2x-3$$)^{3/2}$$] \Big|_2^6$

Question 10

Which substitution is most appropriate for evaluating $$\int \sqrt{\cot x} \csc^2$ x dx$$?

Accepted Answer

$$u = \cot x$$

Question 11

Evaluate $$\int \sec^4$ x 	an x dx$$ using $$u-$$substitution.

Accepted Answer

Question 12

A biologist models population growth with $$P(x) = x^3$ \sqrt{x^2$+1}. $$Which substitution will help integrate $$\int x^3$ \sqrt{x^2$+1} dx$$?

Accepted Answer

$$u = x^2$ + 1$$

Question 13

Which of the following is the correct evaluation of $$\$$int_0$$^{\pi} \frac{\cos x}{\sqrt{1+\sin x}} dx$$?

Accepted Answer

$$2(\sqrt{2} - 1)$$

Question 14

A physicist needs to evaluate $$\int \frac{dx}{\sqrt{1+\sqrt{1+x}}}$$ for a quantum mechanics problem. Which substitution should be used first?

Accepted Answer

$$u = \sqrt{1+x}$$

Question 15

Which substitution is most effective for evaluating $$\$$int_0$$^{\pi/2} \sin^4$ x \cos x dx$$?

Accepted Answer

$$u = \sin x$$

Question 16

A chemist is analyzing a reaction rate modeled by $$R(x) = x^2$ (1-x$$)^{1/2}$$. What is the value of $$\$$int_0^1$$ $$x^2$$ $$(1-x)^{1/2} dx$?

Accepted Answer

$$\frac{4}{15}$$

Question 17

Which substitution is most appropriate for evaluating $$\int \sec(3x) 	an(3x) dx$$?

Accepted Answer

$$u = 3x$$

Question 18

A mathematician wants to evaluate $$\int \sin(2x) dx$$ using $$u-$$substitution. What is the correct answer?

Accepted Answer

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

Section 5.5: U-Substitution in Integration

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