Question 1

Which substitution is most appropriate to evaluate the integral $$\int \cos^2$(	heta) \sin(	heta) d	heta$$?

Accepted Answer

$$u = \sin(	heta)$$

Question 2

Evaluate $$\int (2x-3)^4$ dx$$ using the substitution $$u = 2x-3. $$What is the correct antiderivative?

Accepted Answer

$$\frac{1}{10}(2x-3)^5$ + C$$

Question 3

Which of the following integrals can be solved by the substitution $$u = x^2$ + 1$$?

Accepted Answer

$$\int x \sqrt{x^2$ + 1} dx$$

Question 4

Suppose you want to evaluate $$\$$int_0$$^1 x e$$^{x^2}$$ dx$$ using substitution. What is the best choice for $$u$$?

Accepted Answer

$$u = x^2$

Question 5

Evaluate $$\int \frac{1}{\sqrt{1-x^2$}} dx$$ using substitution. What is the result?

Accepted Answer

$$\arcsin(x) + C$$

Question 6

Which substitution would you use to evaluate $$\int \frac{x}{\sqrt{x^2$+4}} dx$$?

Accepted Answer

$$u = x^2$ + 4$$

Question 7

A physicist models the velocity of a particle as $$v(t) = t \cos(t^2$). $$What substitution should be used to find the displacement $$\$$int_0$$^1 t \cos(t^2$) dt$$?

Accepted Answer

$$u = t^2$

Question 8

Evaluate $$\$$int_0$$^{\pi/2} \sin(x) \cos(x) dx$$ using substitution.

Accepted Answer

$$\frac{1}{2}$$

Question 9

Which of the following is the correct substitution to evaluate $$\int \frac{1}{1+x^2$} dx$$?

Accepted Answer

$$u = x^2$

Question 10

A biologist models the growth of a population with $$P(t) = e$$^{2t}$$. To find the total population from t=0$ to t=1$, which substitution is most useful for $$\$$int_0^1$$ $$e^{2t} dt$?

Accepted Answer

$$u = 2t$$

Question 11

Evaluate $$\int \sin^3$(x) \cos(x) dx$$ using substitution.

Accepted Answer

$$\frac{1}{4} \sin^4$(x) + C$$

Question 12

Which substitution is most appropriate for $$\int x^2$ \sqrt{x^3$ + 1} dx$$?

Accepted Answer

$$u = x^3$ + 1$$

Question 13

Evaluate $$\$$int_0$$^1 2x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

$$e - 1$$

Question 14

Which substitution would you use to evaluate $$\int \frac{1}{\sqrt{4-x^2$}} dx$$?

Accepted Answer

$$x = 2 \sin(	heta)$$

Question 15

A chemist is analyzing a reaction rate modeled by $$r(t) = \frac{1}{1+t^2$}. $$What is the total change in concentration from $$t=0 to$$ $$t=1$$?

Accepted Answer

$$\arctan(1) - \arctan(0)$$

Question 16

Evaluate $$\$$int_0$$^\pi \sin(x) dx$$ using substitution.

Accepted Answer

$$2$$

Question 17

Which substitution is most appropriate for $$\int \frac{2x}{x^2$ + 1} dx$$?

Accepted Answer

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

Question 18

A medical researcher models the concentration of a drug as $$C(t) = \frac{1}{\sqrt{t}}. $$What substitution is most useful for $$\int t$$^{-1/2}$$ dt$$?

Accepted Answer

$$u = t$$^{1/2}$$

Question 19

Evaluate $$\$$int_0$$^1 x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

$$\frac{1}{2}(e^1$ - 1)$$

Techniques of Integration: Substitution and Trigonometric Integrals

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