Question 1

Evaluate the integral.&#10;

Accepted Answer

D

Question 2

Evaluate the integral.&#10;

Accepted Answer

This integral can be solved using integration by parts, where $u = x$ and $dv = 14\sin(x)dx$. The formula for integration by parts is $\int u dv = uv - \int v du$. Applying this, we get $14x(-\cos(x)) - \int -14\cos(x) dx = -14x\cos(x) + 14\sin(x) + C$, which matches choice A after rearranging the terms.

Question 3

Solve the problem.&#10;

Accepted Answer

The volume generated by revolving the curve $y = \cos 5x$ about the x-axis from $0$ to $\pi / 12$ can be found using the formula for the volume of revolution, $V = \pi \int_{a}^{b} [f(x)]^2 dx$. Substituting $f(x) = \cos 5x$ and integrating from $0$ to $\pi / 12$ gives the correct answer as $\frac { \pi ^ { 2 } } { 24 } + \frac { \pi } { 8 }$.

Question 4

Evaluate the integral.&#10;

Accepted Answer

The integral can be solved by completing the square in the denominator to get $x^2 + 6x + 25 = (x+3)^2 + 16$. Then, using the substitution $u = \frac{x+3}{4}$, the integral becomes $\frac{1}{4} \tan^{-1}(u) + C$, which matches choice A.

Question 5

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 6

Evaluate the integral.&#10;

Accepted Answer

This integral requires the use of integration by parts, where $u = x$ and $dv = -6\cos(9x)dx$. After integrating $dv$ and differentiating $u$, we get $du = dx$ and $v = -\frac{2}{3}\sin(9x)$. Applying the integration by parts formula $\int u dv = uv - \int v du$, we obtain $-\frac{2}{3}x\sin(9x) - \int -\frac{2}{3}\sin(9x)dx$. Integrating the second term gives $-\frac{2}{27}\cos(9x)$, leading to the final answer $-\frac{2}{27}\cos(9x) - \frac{2}{3}x\sin(9x) + C$.

Question 7

Evaluate the integral.&#10;

Accepted Answer

This integral can be solved by recognizing it as a form suitable for substitution. Let $u = t^5 + 4$, then $du = 5t^4 dt$. The integral becomes $\int \frac{1}{5} \ln(u) du$, which integrates to $\frac{1}{5} \cdot \frac{u \ln(u) - u}{1} + C$. Simplifying, we get $\frac{1}{5} (u \ln(u) - u) + C$. However, the correct answer is obtained by recognizing the integral as a form that leads to a result involving $\ln(u)^2$, which is a common mistake. The correct method involves recognizing that the integral directly does not lead to a simple antiderivative involving $\ln(u)^2$. The mistake in the explanation points towards a misunderstanding of the integral's solution. The correct approach involves recognizing the integral setup and applying integration techniques correctly, but the explanation provided mistakenly suggests a solution that does not match the integral's form. The correct answer involves recognizing the integral as a direct application of a specific integration technique, but the explanation erroneously suggests a common mistake.

Question 8

Evaluate the integral.&#10;

Accepted Answer

The integral can be solved using substitution. Let $ u = \cos \theta $, then $ du = -\sin \theta d\theta $. The integral becomes $-\int u^4 du$, which evaluates to $-\frac{1}{5} u^5 + C$. Substituting back, we get $-\frac{1}{5} \cos^5 \theta + C$. However, considering the factor of $\sin 2\theta = 2\sin\theta\cos\theta$, the correct integration results in $-\frac{1}{3} \cos^6 \theta + C$.

Question 9

Evaluate the integral.&#10;

Accepted Answer

This integral can be solved using integration by parts, where $u = x$ and $dv = \csc^2 3x dx$. The derivative of $u$ is $du = dx$, and the integral of $dv$ is $v = -\frac{1}{3}\cot 3x$. Applying the integration by parts formula, $\int u dv = uv - \int v du$, gives $-\frac{1}{3}x\cot 3x + \frac{1}{9}\ln|\sin 3x| + C$.

Question 10

Evaluate the integral.&#10;

Accepted Answer

To solve the integral $\int \frac { \sec \theta } { \sec \theta - \tan \theta } d \theta$, we can use a substitution method where we let $u = \sec \theta - \tan \theta$. Differentiating both sides with respect to $\theta$ gives $du = (\sec \theta \tan \theta - \sec^2 \theta) d\theta$. However, a more straightforward approach without explicit substitution involves recognizing that the denominator can be rewritten as a single expression in terms of sine and cosine, and then the integral can be approached directly or through trigonometric identities. The correct answer involves recognizing that the integral simplifies to a form where the antiderivative directly leads to $\tan \theta + \sec \theta + C$, which is a result of integrating the given function properly and applying trigonometric identities.

Question 11

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 12

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 13

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 14

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 15

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 16

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 17

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 18

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 19

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 20

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 21

Evaluate the integral.&#10; &#10;A) 20.1&#10;B) 6.70&#10;C) 26.1&#10;D) 4.74

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 20.1&#10;B) 6.70&#10;C) 26.1&#10;D)...

Question 22

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 23

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 24

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 25

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 26

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 27

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 28

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 29

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 30

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 31

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 32

Evaluate the integral by using a substitution prior to integration by parts.&#10; &#10;A) -2.27&#10;B) 0.39&#10;C) -1.33&#10;D) -0.94

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 33

Evaluate the integral.&#10; &#10;A) 187.01&#10;B) -61.21&#10;C) 219.01&#10;D) 190.21

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 187.01&#10;B) -61.21&#10;C) 219.01&#10;D)...

Question 34

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 35

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 36

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 37

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 38

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 39

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 40

Evaluate the integral.&#10; &#10;A) 8.51&#10;B) -3.49&#10;C) 14.4&#10;D) 4.51

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 8.51&#10;B) -3.49&#10;C) 14.4&#10;D)...

Question 41

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 42

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 43

Use integration by parts to establish a reduction formula for the integral.&#10;

Accepted Answer

The answer of Use integration by parts to establish a...

Question 44

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 45

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 46

Evaluate the integral. The integral may not require integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral. The integral may not...

Question 47

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 48

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 49

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 50

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 51

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 52

Use integration by parts to establish a reduction formula for the integral.&#10;

Accepted Answer

The answer of Use integration by parts to establish a...

Question 53

Evaluate the integral by using a substitution prior to integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral by using a substitution...

Question 54

Evaluate the integral. The integral may not require integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral. The integral may not...

Question 55

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 56

Solve the problem.&#10;Find the area between y = ln x and the x-axis from x = 1 to x = 3.

Accepted Answer

The answer of Solve the problem.&#10;Find the area between y...

Question 57

Evaluate the integral. The integral may not require integration by parts.&#10;

Accepted Answer

The answer of Evaluate the integral. The integral may not...

Question 58

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 59

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 60

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 61

Use integration by parts to establish a reduction formula for the integral.&#10;

Accepted Answer

The answer of Use integration by parts to establish a...

Question 62

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 63

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 64

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Deck 9: Techniques of Integration