Question 1

Evaluate the integral.&#10;

Accepted Answer

The integral can be evaluated using integration by parts or recognizing it as a Laplace transform. The result is $\frac{1}{5}$.

Question 2

Evaluate the integral.&#10;

Accepted Answer

A

Question 3

Evaluate the integral.&#10;

Accepted Answer

We use integration by parts with $u = \ln(4x)$ and $dv = x^3 \, dx$, so $du = \frac{1}{x} \, dx$ and $v = \frac{1}{4}x^4$. Then
$$\int x^3 \ln(4x) \,dx = \frac{1}{4}x^4 \ln(4x) - \int \frac{1}{4}x^3 \, dx.$$ Integrating the remaining integral gives
$$\int x^3 \ln(4x) \,dx = \frac{1}{4}x^4 \ln(4x) - \frac{1}{16}x^4 + C.$$ The only answer choice that matches this is $\boxed{	extbf{(B)}}$.

Question 4

Evaluate the integral.&#10;

Accepted Answer

To solve the integral $\int x \csc^2 3x \, dx$, we use integration by parts, where $u = x$ and $dv = \csc^2 3x \, dx$. This gives $du = dx$ and $v = -\frac{1}{3}\cot 3x$, since the integral of $\csc^2 x$ is $-\cot x$. Applying the integration by parts formula $\int u \, dv = uv - \int v \, du$, we get $-\frac{1}{3} x \cot 3x - \int -\frac{1}{3} \cot 3x \, dx$. The integral of $\cot x$ is $\ln |\sin x|$, so the final answer is $-\frac{1}{3} x \cot 3x + \frac{1}{9} \ln |\sin 3x| + C$, which matches choice B.

Question 5

Evaluate the integral.&#10;

Accepted Answer

This integral can be solved using integration by parts, where $u = \ln(14x)$ and $dv = (2x - 1)dx$. After integrating $dv$ to get $v = x^2 - x$ and differentiating $u$ to get $du = \frac{1}{x}dx$, applying the integration by parts formula $\int u dv = uv - \int v du$ gives $(x^2 - x)\ln(14x) - \int(x^2 - x)\frac{1}{x}dx$. Simplifying the integral part gives $-\frac{x^2}{2} + x$, leading to the final answer of $(x^2 - x)\ln(14x) - \frac{x^2}{2} + x + C$.

Question 6

Evaluate the integral.&#10;

Accepted Answer

This integral can be solved using integration by parts or the tabular method, repeatedly applying the technique to reduce the power of $y$ each time. The correct result involves a combination of $y^3$, $y^2$, $y$, and constant terms, all multiplied by $e^{-3y}$, with coefficients that ensure each differentiation step according to the power of $y$ is accounted for correctly. Option A correctly represents this process and its result, including the negative sign outside which comes from integrating $e^{-3y}$.

Question 7

Evaluate the integral.&#10;

Accepted Answer

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

Question 8

Evaluate the integral.&#10;

Accepted Answer

Use integration by parts. Let $u = \cos^{-1} x$ and $dv = dx$, then we have
$$du = -\frac{1}{\sqrt{1-x^2}}dx \quad \text{and} \quad v = x.$$
Using the formula for integration by parts, we get
\begin{align*}
\int \cos ^ { - 1 } x d x &= uv - \int v du \
&= x\cos^{-1}x - \int x\frac{1}{\sqrt{1-x^2}}dx \
&= x\cos^{-1}x - \left[-\sqrt{1-x^2}\right] + C \
&= x\cos^{-1}x + \sqrt{1-x^2} + C.
\end{align*}
Therefore, the correct choice is $\boxed{\text{(B)}}.$

Question 9

Evaluate the integral.&#10;

Accepted Answer

We can begin by integrating by parts, with $u=\ln x$ and $dv=7x dx$. Then, we get:
$$\int 7x \ln x d x = \frac{7x^2 \ln x}{2} - \int \frac{7x}{2} dx$$
Integrating the second term gives us:
$$\frac{7x^2 \ln x}{2} - \frac{7}{4} x^2$$
Evaluating from 2 to 4:
$$\frac{7(4)^2 \ln 4}{2} - \frac{7}{4} (4)^2 - \frac{7(2)^2 \ln 2}{2} + \frac{7}{4} (2)^2 = 46.9$$
Therefore, the answer is D.

Question 10

Evaluate the integral.&#10;

Accepted Answer

We can use integration by parts with $u=2x$ and $dv=e^xdx$ to get $$\int 2 x e ^ { x } d x = 2xe^x- \int 2e^xdx=2xe^x-2e^x+C.$$ Hence, the correct choice is (C).

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

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#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

Solve the problem.&#10;

Accepted Answer

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

Question 22

Solve the problem.&#10;

Accepted Answer

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

Question 23

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 24

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

Accepted Answer

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

Question 27

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 28

Solve the problem.&#10;

Accepted Answer

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

Question 29

Solve the problem.&#10;

Accepted Answer

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

Question 30

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 31

Solve the problem.&#10;

Accepted Answer

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

Question 32

Solve the problem.&#10;

Accepted Answer

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

Question 33

Solve the problem.&#10;

Accepted Answer

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

Question 34

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 35

Solve the problem.&#10;

Accepted Answer

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

Question 36

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 37

Solve the problem.&#10;

Accepted Answer

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

Question 38

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 39

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 40

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 41

Solve the problem.&#10;

Accepted Answer

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

Question 42

Evaluate the integral.&#10;

Accepted Answer

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

Question 43

Evaluate the integral.&#10;

Accepted Answer

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

Question 44

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 45

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 46

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 47

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 48

Solve the problem.&#10;

Accepted Answer

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

Question 49

Evaluate the integral.&#10;

Accepted Answer

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

Question 50

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 51

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 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

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 54

Solve the problem.&#10; &#10;A) 0.13 gallons per hour&#10;B) 3.07 gallons per hour&#10;C) 3.11 gallons per hour&#10;D) 3.33 gallons per hour

Accepted Answer

The answer of Solve the problem.&#10; &#10;A) 0.13 gallons per...

Question 55

Solve the problem.&#10;

Accepted Answer

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

Question 56

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 57

Solve the problem.&#10;

Accepted Answer

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

Question 58

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 59

Evaluate the integral.&#10;

Accepted Answer

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

Question 60

Solve the problem.&#10;

Accepted Answer

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

Question 61

Evaluate the integral.&#10;

Accepted Answer

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

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 8: Techniques of Integration