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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