Question 1

Evaluate the integral.&#10;

Accepted Answer

We can start by letting $u = 5 + \sec x$. Then $du = \sec x 	an x dx$, which we can use to substitute in the integral:
\begin{align*}
\int \frac{\sec x 	an x}{5+\sec x}dx &= \frac{1}{5} \int \frac{du}{u} \
&= \frac{1}{5} \ln|u| + C \
&= \frac{1}{5} \ln|5+\sec x| + C.
\end{align*}
Therefore, the correct choice is (C).

Question 2

Evaluate the integral.&#10;

Accepted Answer

The integral can be solved by substitution. Let $ u = \sqrt{7x} $, then $ du = \frac{7}{2\sqrt{x}} dx $. The integral becomes $\int 3e^u du = 3e^u + C$. Substituting back, we get $3e^{\sqrt{7x}} + C$, which matches option D.

Question 3

Evaluate the integral.&#10;

Accepted Answer

We can use substitution method $u=x^4$, so $du=4x^3 dx$. Then the integral becomes: $$\int x^3e^{-x^4}dx=\frac{1}{4}\int e^{-u}du=-\frac{1}{4}e^{-u}+C=-\frac{1}{4}e^{-x^4}+C$$ So, the correct answer is (C).

Question 4

Evaluate the integral.&#10;

Accepted Answer

We can start by making a substitution $u=1+4\sin x$:
\begin{align*}
\int \frac{\cos x}{1+4\sin x}dx &=\frac{1}{4}\int \frac{d(1+4\sin x)}{1+4\sin x}\cdot\cos x\
&=\frac{1}{4}\ln|1+4\sin x|+C
\end{align*}
Therefore, the answer is C.

Question 5

Evaluate the integral.&#10;

Accepted Answer

First, we use the substitution $u = 	an 2x$ and $du = 2\sec^2 2x dx$, which turns the integral into
$$\frac{1}{2} \int_0^{1} (1+e^u) du$$
Using the bounds of integration, this becomes
$$\frac{1}{2} \int_0^{	an(\pi/4)} (1+e^u) du = \frac{1}{2} \left[ u + e^u ight]_0^{	an(\pi/4)}$$
$$= \frac{1}{2} \left[ 	an 2x + e^{	an 2x} ight]_0^{\pi/8}$$
Substituting the limits of integration, this becomes
$$\frac{1}{2} \left[ 	an \frac{\pi}{4} - 	an 0 + e^{	an 0} - e^{-	an{\pi}/{4}} ight]$$
Simplifying this expression, we get
$$\frac{1}{2} \left[1 + 1 - \frac{1}{e^{1/2}} ight] = \frac{e}{2} - \frac{1}{2e^{1/2}}$$
Since $e^{1/2} > 1$ then $1/e^{1/2} < 1$ and so $\frac{1}{2e^{1/2}} < \frac{1}{2}$. Therefore, $\frac{e}{2} - \frac{1}{2e^{1/2}} < \frac{e}{2}$. So, the answer is $\boxed{	extbf{(C)}}$.

Question 6

Evaluate the integral.&#10;

Accepted Answer

The integral of $\tan \frac{x}{5}$ can be solved by using the substitution method. Let $u = \frac{x}{5}$, which implies $du = \frac{1}{5}dx$, or $dx = 5du$. Substituting these into the integral gives $\int \tan(u) \cdot 5du$, which simplifies to $5 \int \tan(u) du$. The integral of $\tan(u)$ is $\ln|\sec(u)| + C$, so the integral becomes $5\ln|\sec(u)| + C$. Substituting back for $u = \frac{x}{5}$, we get $5\ln|\sec(\frac{x}{5})| + C$. Evaluating this from $0$ to $5\pi/4$ gives $5[\ln|\sec(5\pi/4/5)| - \ln|\sec(0)|]$, which simplifies to $5[\ln|\sec(\pi/4)| - \ln|\sec(0)|]$. Since $\sec(\pi/4) = \sqrt{2}$ and $\sec(0) = 1$, the result is $5[\ln|\sqrt{2}| - \ln|1|] = 5\ln\sqrt{2}$. Using the property of logarithms that $\ln a^b = b\ln a$, this becomes $5\frac{1}{2}\ln 2 = \frac{5\ln 2}{2}$.

Question 7

Evaluate the integral.&#10; &#10;A) -2 ln 2&#10;B) 2 ln 3&#10;C) 4 ln 2&#10;D) 2 ln 2

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) -2 ln 2&#10;B)...

Question 8

Evaluate the integral.&#10;

Accepted Answer

We can start by factoring $3$ out of the denominator, giving us $\frac{8}{3} \cdot \frac{1}{(3+x/3)}$. Now we can use the substitution $u=3+x/3$ and $du = \frac{1}{3}dx$. This gives us:
\begin{align*}
\int \frac { 8 d x } { 9 + 3 x } &=  \int \frac{8}{3} \cdot \frac{1}{(3+x/3)} dx \
&= 8 \int \frac{1}{u} \cdot \frac{1}{3}du \
&=  \frac{8}{3} \ln|u| + C \
&= \frac{8}{3} \ln|3+x/3| + C \
&= \frac{8}{3} \ln|-9-3x| + C \qquad (	ext{since } u=3+x/3 \implies 3+x/3=-9-3x)
\end{align*}
Therefore, the correct choice is $\boxed{	ext{D}}$.

Question 9

Evaluate the integral.&#10;

Accepted Answer

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

Question 10

Evaluate the integral.&#10;

Accepted Answer

The integral can be solved by recognizing it as a form suitable for substitution. Let $u = 4 + 6\ln x$, then $du = \frac{6}{x}dx$. The integral becomes $\int \frac{1}{6} \frac{du}{u}$, which evaluates to $\frac{1}{6}\ln|u| + C$, or $\frac{1}{6}\ln|4 + 6\ln x| + C$.

Question 11

Evaluate the integral.&#10; &#10;A) 11&#10;B) -3&#10;C) 3&#10;D) 14

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 11&#10;B) -3&#10;C) 3&#10;D)...

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; &#10;A) -28 ln 3&#10;B) 28 ln 3&#10;C) 14 ln 3&#10;D) -14 ln 3

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) -28 ln 3&#10;B)...

Question 19

Evaluate the integral.&#10; &#10;A) 1 + cos 1&#10;B) -1&#10;C) 1&#10;D) 1 - cos 1

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 1 + cos...

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

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.&#10;

Accepted Answer

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

Question 26

Evaluate the integral.&#10;

Accepted Answer

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

Question 27

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#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

Solve the initial value problem.&#10;

Accepted Answer

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

Question 32

Evaluate the integral.&#10;

Accepted Answer

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

Question 33

Solve the initial value problem.&#10;

Accepted Answer

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

Question 34

Solve the initial value problem.&#10;

Accepted Answer

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

Question 35

Solve the initial value problem.&#10;

Accepted Answer

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

Question 36

Evaluate the integral.&#10;

Accepted Answer

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

Question 37

Solve the initial value problem.&#10;

Accepted Answer

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

Question 38

Evaluate the integral.&#10;

Accepted Answer

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

Question 39

Solve the initial value problem.&#10;

Accepted Answer

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

Question 40

Solve the initial value problem.&#10;

Accepted Answer

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

Question 41

Solve the differential equation.&#10;

Accepted Answer

The answer of Solve the differential equation.&#10; ...

Question 42

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 43

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) No&#10;B) Yes

Accepted Answer

The answer of Determine if the given function y =...

Question 44

Solve the problem.&#10;

Accepted Answer

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

Question 45

Solve the problem.&#10;

Accepted Answer

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

Question 46

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 47

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 48

Solve the problem.&#10;

Accepted Answer

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

Question 49

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 50

Solve the differential equation.&#10;

Accepted Answer

The answer of Solve the differential equation.&#10; ...

Question 51

Solve the problem.&#10;

Accepted Answer

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

Question 52

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 53

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 54

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 55

Solve the differential equation.&#10;

Accepted Answer

The answer of Solve the differential equation.&#10; ...

Question 56

Solve the differential equation.&#10;

Accepted Answer

The answer of Solve the differential equation.&#10; ...

Question 57

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 58

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 59

Determine if the given function y = f(x) is a solution of the accompanying differential equation.&#10; &#10;A) Yes&#10;B) No

Accepted Answer

The answer of Determine if the given function y =...

Question 60

Solve the differential equation.&#10;

Accepted Answer

The answer of Solve the differential equation.&#10; ...

Question 61

11eb49f8_dfea_8e74_b71a_ffc88850e7d2_TB6591_0011eb49f8_dfea_dc99_b71a_db69758d0347_TB6591_00

Accepted Answer

The answer of 11eb49f8_dfea_8e74_b71a_ffc88850e7d2_TB6591_0011eb49f8_dfea_dc99_b71a_db69758d0347_TB6591_00...

Question 62

Solve the problem.&#10; &#10;A) 800 years&#10;B) 1400 years&#10;C) 2308 years&#10;D) 1600 years

Accepted Answer

The answer of Solve the problem.&#10; &#10;A) 800 years&#10;B) 1400...

Question 63

Solve the differential equation.&#10;

Accepted Answer

The answer of Solve the differential equation.&#10; ...

Question 64

11eb49f8_dfea_8e74_b71a_ffc88850e7d2_TB6591_0011eb49f8_dfea_b588_b71a_a12520f2543b_TB6591_00

Accepted Answer

The answer of 11eb49f8_dfea_8e74_b71a_ffc88850e7d2_TB6591_0011eb49f8_dfea_b588_b71a_a12520f2543b_TB6591_00...

Deck 8: Integrals and Transcendental Functions