Deck 7: Integration

Question 1

Integrate $\int(\tan 7 x+\cos 7 x) d x$ .

A) $\frac{\ln |\cos 7 x|}{7}-\frac{\sin 7 x}{7}+C$
B) $-\frac{\ln |\cos 7 x|}{7}+\frac{\sin 7 x}{7}+C$
C) $\frac{\cot 7 x}{7}-\frac{\sin 7 x}{7}+C$
D) $-\frac{\cot 7 x}{7}+\frac{\sin 7 x}{7}+C$

Answer 1

$-\frac{\ln |\cos 7 x|}{7}+\frac{\sin 7 x}{7}+C$

Question 2

Find an antiderivative of $x^{2}-\frac{6}{x}+\frac{8}{x^{3}}$ .

A) $\frac{x^{3}}{3}-\frac{12}{x^{2}}-\frac{32}{x^{4}}+C$
B) $\frac{x^{3}}{3}-\frac{6}{x}+\frac{8}{x^{2}}+C$
C) $\frac{x^{3}}{3}-6 \ln |x|+\frac{8}{x^{2}}+C$
D) $\frac{x^{3}}{3}-6 \ln |x|-\frac{4}{x^{2}}+C$

Answer 2

$\frac{x^{3}}{3}-6 \ln |x|-\frac{4}{x^{2}}+C$

Question 3

Find the area between $f(x)=4 x e^{-x^{2}}$ and g(x)= x for x $\ge$ 0.Round to 3 decimal places.

Answer 3

0.807

Question 4

Compute $\int_{-R}^{R} \frac{\sin x}{9+x^{4}} d x$ .

A) $\frac{2 \cos R}{9 R+R^{5}}$
B) $\frac{3 \cos \left(9+R^{4}\right)}{R^{3}}$
C) $\frac{1}{3\left(9+R^{4}\right)}$
D)0

Question 5

Use the Fundamental Theorem to evaluate the definite integral .Reduce fractions and leave them in the form "A/B".

Question 6

Find $\int \frac{5 x}{\sqrt{4-x^{2}}} d x$

A) $-\frac{10}{3\left(4-x^{2}\right)^{3 / 2}}+C$
B) $\frac{10}{3\left(4-x^{2}\right)^{3 / 2}}+C$
C) $5 \sqrt{4-x^{2}}+C$
D) $-5 \sqrt{4-x^{2}}+C$

Question 7

Find $\int e^{-t / 4} \sin 4 t d t$ .

A) $-\frac{16}{257} e^{-t / 4}\left(\frac{1}{4} \sin 4 t+4 \cos 4 t\right)+C$
B) $\frac{16}{257} e^{-t / 4}\left(\frac{1}{4} \sin 4 t-4 \cos 4 t\right)+C$
C) $-\frac{1}{32} e^{-t / 4}\left(\frac{1}{4} \sin 4 t+4 \cos 4 t\right)+C$
D) $\frac{1}{32} e^{-t / 4}\left(\frac{1}{4} \sin 4 t-4 \cos 4 t\right)+C$

Question 8

Find an antiderivative of $(x+\sqrt{\sin (3 x+9)})(x-\sqrt{\sin (3 x+9)})$ .

A) $\frac{x^{3}}{3}+\frac{1}{3} \cos (3 x+9)+C$
B) $\frac{x^{3}}{3}-\frac{1}{3} \cos (3 x+9)+C$
C) $\frac{x^{3}}{3}+\frac{1}{3 x+9} \cos (3 x+9)+C$
D) $\frac{x^{3}}{3}-\frac{1}{3 x+9} \cos (3 x+9)+C$

Question 9

Evaluate $\int 6 x^{7} d x$ .

A) $\frac{3}{4} x^{8}+C$
B) $x^{8}+C$
C) $\frac{6}{7} x^{8}+C$
D) $\frac{7}{8} x^{8}+C$

Question 10

Find $\int(\ln x)^{2} d x$ .Hint: Integrate by parts.

A) $x(\ln x)^{2}+2 x \ln x-2 x+C$
B) $x(\ln x)^{2}-2 x \ln x+2 x+C$
C) $x(\ln x)^{2}-2 x \ln x+C$
D) $x(\ln x)^{2}+2 x \ln x+C$

Question 11

Evaluate $\int 8 e^{x} \cos \left(8 e^{x}\right) d x$ .

A) $\frac{\sin \left(8 e^{x}\right)}{e^{x}}+C$
B) $\frac{1}{9} \sin \left(8 e^{x}\right)+C$
C) $\sin \left(8 e^{x}\right)+C$
D) $\frac{8 e^{x+1}}{x+1} \sin \left(8 e^{x}\right)+C$

Question 12

Evaluate $\int \frac{x^{2}-x+5}{x} d x$ .

A) $\ln |x|\left(\frac{x^{3}}{3}-\frac{x^{2}}{2}+5 x\right)+C$
B) $\frac{x^{2}}{2}-x+5 \ln |x|+C$
C) $\frac{2 x}{3}+\frac{10}{x}+C$
D) $\frac{x^{2}}{2}-x+\frac{10}{x^{2}}+C$

Question 13

Suppose $\int_{0}^{2} f(t)=a$ , where a is a constant.Calculate $\int_{0}^{2} 5 f(2-t) d t$ .

A) $-10 a$
B) $10 a$
C) $-5 a$
D) $5 a$

Question 14

Calculate the area between the curve and the x-axis between and .Round your answer to 2 decimal places.

Question 15

Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of psi per second at time t in seconds.At what rate, in psi/sec, is pressure decreasing at 10 seconds? Round to 2 decimal places.

Question 16

Find $\int 3 y\left(y^{2}+2\right)^{3} d y$

A) $\frac{3}{8 y}\left(y^{2}+2\right)^{4}+C$
B) $\frac{3 y^{2}}{8}\left(y^{2}+2\right)^{4}+C$
C) $\frac{3}{8}\left(y^{2}+2\right)^{4}+C$
D) $\frac{3}{4}\left(y^{2}+2\right)^{4}+C$

Question 17

Compute $\int_{0}^{R} \frac{1}{(3+x)^{2}} d x$ .

A) $\frac{1}{3+R}$
B) $\frac{1}{3}-\frac{1}{3+R}$
C) $\frac{1}{3(3+R)^{3}}$
D) $\frac{3}{R^{3}}-\frac{3}{(3+R)^{3}}$

Question 18

Integrate $\int \cos ^{2} 3 x \sin 3 x d x$ .

A) $\frac{\cos ^{3} 3 x}{9}+C$
B) $-\frac{\cos ^{3} 3 x}{9}+C$
C) $\frac{\cos ^{3} 3 x}{3}+C$
D) $-\frac{\cos ^{3} 3 x}{27}+C$

Question 19

Suppose $\int_{0}^{7} f(t) d t=a$ , where a is a constant.Calculate $\int_{0}^{1} f(7 t) d t$ .

A)a
B)7a
C) $\frac{a}{7}$
D) $a-7$

Question 20

Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of psi per second at time t in seconds.By how many total psi has the pressure decreased during the first minute? Round to 2 decimal places.

Question 21

Integrate $\int \frac{x^{4}+5}{x^{3}} d x$ .

A) $\frac{x^{2}}{2}-\frac{5}{2 x^{2}}+C$
B) $\frac{x^{2}}{2}+\frac{5}{2 x^{2}}+C$
C) $\frac{2}{5} \cdot \frac{x^{5}+5 x}{x^{2}}+C$
D) $\frac{x^{5}+5 x}{x^{2}}+C$

Question 22

Calculate $\int \frac{d t}{3+\sqrt{t}}$ .

A) $2(3+\sqrt{t})+6 \ln (3+\sqrt{t})+C$
B) $2(3+\sqrt{t})-6 \ln (3+\sqrt{t})+C$
C) $\frac{\sqrt{t} \ln (3+\sqrt{t})}{2}+C$
D) $-\frac{\sqrt{t} \ln (3+\sqrt{t})}{2}+C$

Question 23

Find $\int\left(\frac{1}{x+1}-\frac{1}{(x+1)^{2}}\right) d x$ .

A) $\ln |x+1|-\frac{3}{(x+1)^{3}}+C$
B) $\ln |x+1|+\frac{3}{(x+1)^{3}}+C$
C) $\ln |x+1|-\frac{1}{x+1}+C$
D) $\ln |x+1|+\frac{1}{x+1}+C$

Question 24

Use the table of antiderivatives to determine if the following statement is true. $\int \frac{d x}{\sqrt{x^{2}-2 x+2}}=\ln \left|x+\sqrt{x^{2}-2 x+2}\right|+C$

Question 25

Calculate $\int z e^{5 z^{2}+8} d z$ .

A) $\frac{z}{5} e^{5 z^{2}+8}+C$
B) $\frac{z}{10} e^{5 z^{2}+8}+C$
C) $\frac{1}{5} e^{5 z^{2}+8}+C$
D) $\frac{1}{10} e^{z^{2}+8}+C$

Question 26

$\int \theta^{2} \cos (a \theta) d \theta=\frac{\theta^{2}}{a} \sin (a \theta)+\frac{2 \theta}{a^{2}} \cos (a \theta)+C$ , where a is a constant.

Question 27

For $f(x)=x^{2} e^{10 x}$ , find a function $F(x)$ such that $F^{\prime}(x)=f(x)$ and $F(0)=0$ .

A) $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x+\frac{1}{500}\right)-\frac{1}{500}$
B) $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x+\frac{1}{500}\right)$
C) $e^{10 x}\left(\frac{1}{10} x^{2}-\frac{1}{50} x\right)$
D) $e^{10 x}\left(\frac{x^{3}}{30}\right)$

Question 28

$\int \ln x d x=x \ln x-x+C$ .

Question 29

Calculate $\int \sec ^{2} \theta d \theta$

A) $\frac{3 \sin \theta}{\cos ^{3} \theta}+C$
B) $-\frac{1}{\cos \theta}+C$
C) $\tan \theta+C$
D) $\tan ^{2} \theta+C$

Question 30

For $f(x)=x \sin (4 x)$ , find a function $F(x)$ such that $F^{\prime}(x)=f(x)$ and $F(0)=0$ .

A) $\frac{1}{4} x \cos (4 x)-\frac{1}{4} \sin (4 x)$
B) $-\frac{1}{4} x \cos (4 x)+\frac{1}{4} \sin (4 x)$
C) $\frac{1}{4} x \cos (4 x)-\frac{1}{16} \sin (4 x)$
D) $-\frac{1}{4} x \cos (4 x)+\frac{1}{16} \sin (4 x)$

Question 31

Calculate $\int \frac{d x}{(b+a x)^{8}}$ , where a and b are constants.

A) $\frac{9}{a(b+\alpha x)^{9}}+C$
B) $\frac{7 a}{(b+a x)^{7}}+C$
C) $-\frac{1}{7 a(b+a x)^{7}}+C$
D) $\frac{1}{7 a(b+ a x)^{7}}+C$

Question 32

Find $\int x \cos 2 x d x$ .

A) $\frac{1}{2} x \sin 2 x+\frac{1}{4} \cos 2 x+C$
B) $\frac{1}{2} x \sin 2 x-\frac{1}{2} \cos 2 x+C$
C) $-\frac{x^{2}}{8} \sin 2 x+C$
D) $\frac{x^{2}}{4} \sin 2 x+C$

Question 33

Integrate .Give an exact answer and one rounded to 3 decimal places.

Question 34

Integrate $\int \sin (3 x) e^{\cos (3 x)} d x$ .

A) $-e^{\cos (3 x)}+C$
B) $e^{\cos (3 x)}+C$
C) $-\frac{1}{3} e^{\cos (3 x)}+C$
D) $\frac{1}{3} e^{\cos (3 x)}+C$

Question 35

Calculate $\int y \sec ^{2} y d y$ .

A) $y \tan y+\ln |\cos y|+C$
B) $y \tan y-\ln |\cos y|+C$
C) $\frac{y^{2}}{2}-\tan y+C$
D) $\frac{y^{2}}{2} \tan y+C$

Question 36

For $f(x)=x^{2}\left(8+x^{3}\right)^{10}$ , find a function $F(x)$ such that $F^{\prime}(x)=f(x)$ and $F(0)=0$ .

A) $\frac{1}{33}\left(\left(8+x^{3}\right)^{11}\right)$
B) $\frac{1}{33}\left(\left(8+x^{3}\right)^{11}-8^{11}\right)$
C) $\frac{1}{99}\left(\left(8+x^{3}\right)^{11}-8^{11}\right)$
D) $\frac{x}{99}\left(8+x^{3}\right)^{11}$

Question 37

Use the table of antiderivatives to determine if the following statement is true. $\int \frac{x^{2}}{x^{2}+5} d x=x-\sqrt{5} \arctan \frac{x}{\sqrt{5}}+C$

Question 38

Integrate $\int \frac{\sqrt[4]{\ln x}}{x} d x$ .

A) $(\ln x)^{5 / 4}+C$
B) $\frac{4}{5}(\ln x)^{5 / 4}+C$
C) $\frac{2(\ln x)^{5 / 4}}{x^{2}}+C$
D) $\frac{\sqrt[5]{\ln x}}{5}+C$

Question 39

Use the table of antiderivatives to determine if the following statement is true. $\int \frac{d x}{x^{2}-10 x+26}=\arctan (x-5)+C$

Question 40

$\int t e^{a t} d t=\frac{1}{a} t e^{a t}-\frac{1}{a^{2}} e^{a t}+C$ , where a is a constant.

Question 41

$\int \frac{3 e^{3 x}}{1+e^{6 x}} d x=\tan ^{-1}\left(e^{3 x}\right)+C$ .

Question 42

Use the table of antiderivatives to determine if the following statement is true. $\int \frac{d t}{\sqrt{4-7 t^{2}}}=\frac{1}{7} \arcsin \left(\frac{t \sqrt{7}}{2}\right)+C$

Question 43

$\int \sin ^{-3} x d x=\frac{-1}{-3} \sin ^{-4} x \cos x+\frac{-3-1}{-3} \int \sin ^{-5} x d x$

Question 44

Find the area of the region bounded by y = 0 and between t = 0 and t = 2.Round to 3 decimal places.

Question 45

The following are some of the values for a function known as the Gudermannian function, G(x).
Use these values to approximate the value of using the trapezoid rule.

Question 46

Which of the following gives the area of the circle $x^{2}+y^{2}=1$ ?

A) $\int_{-1}^{1} \sqrt{1-x^{2}} d x$
B) $4 \int_{-1}^{1} \sqrt{1-x^{2}} d x$
C) $4 \int_{0}^{1} \sqrt{1-x^{2}} d x$
D) $2 \int_{0}^{1} \sqrt{1-x^{2}} d x$

Question 47

Compute .Round to 3 decimal places.

Question 48

Consider the semicircle of radius 4 pictured below.Which of the following could represent the area of the semicircle? Select all that apply.

A) $\int_{-4}^{4} \sqrt{16-x^{2}} d x$
B) $\frac{1}{2} \cdot \pi \cdot 4^{2}$
C) $4 \int_{-1}^{1} \sqrt{16+x^{2}} d x$
D) $\left.\frac{1}{2}\left[x \sqrt{16-x^{2}}+16 \arcsin \left(\frac{x}{4}\right)\right]\right|_{-4} ^{4}$
E) $\left.\frac{2}{4}\left[x \sqrt{16-x^{2}}\right]\right|_{-4} ^{4}$

Question 49

$\int \frac{t+5}{t^{2}+10 t+75} d t=\ln \left|t^{2}+10 t+75\right|+C$ .

Question 50

$\int \cos (6 \theta) d \theta=-\frac{1}{6} \sin (6 \theta)+C$ .

Question 51

Find $\int \cos ^{3} \beta d \beta$ .

A) $\sin \beta-\frac{1}{3} \sin ^{3} \beta+C$
B) $\frac{1}{3} \sin \beta+\frac{2}{3} \sin ^{3} \beta+C$
C) $\sin \beta-\frac{2}{3} \sin ^{3} \beta+C$
D) $\frac{1}{3} \sin \beta+\sin ^{3} \beta+C$

Question 52

The following numbers are the left, right, trapezoidal, and midpoint approximations to , where f(x)is as shown.(Each uses the same number of subdivisions.)
• 0.36735
• 0.39896
• 0.36814
• 0.33575 Which one is the midpoint approximation?

Question 53

Find the area of the region bounded by , x = 0, and x = 2.Round to 3 decimal places.

Question 54

Compute .Round to 3 decimal places.

Question 55

Suppose that as a storm dies down, its rainfall rate (in inches/hour)is given by $y=\frac{1}{0.09+t^{2}}$ for 0 $\le$ t $\le$ 2, where t is the number of hours since the point of heaviest rainfall.What is the average rainfall rate over these two hours? Round your answer to 3 decimal places.

Question 56

Find $\int \frac{4 x+1}{4 x^{2}+4 x} d x$ .

A) $\frac{3}{4} \ln |x|+\frac{1}{4} \ln |x+1|+C$
B) $\frac{1}{4} \ln |x|+\frac{3}{4} \ln |x+1|+C$
C) $\frac{3}{4} \ln |x|+\frac{1}{4} \ln |x+1|+C$ .
D) $\ln |x|+\frac{1}{4} \ln |x+1|+C$

Question 57

$\int \frac{2 d x}{x^{2}+4 x+3}=\ln |x+1|-\ln |x+3|+C$ .

Question 58

Use the table of antiderivatives to determine if the following statement is true. $\int e^{7 x} \cos 8 x d x=\frac{1}{113} e^{7 x}(7 \cos 8 x+8 \sin 8 x)+C$

Question 59

Find $\int \frac{1}{x^{2}+3 x-10} d x$ .

A) $-\frac{1}{10}(\ln |x+5|+\ln |x-2|)+C$
B) $\frac{1}{7}(\ln |x+5|+\ln |x-2|)+C$
C) $\frac{1}{10}(\ln |x+5|-\ln |x-2|)+C$
D) $-\frac{1}{7}(\ln |x+5|-\ln |x-2|)+C$

Question 60

$\int \sin (\sqrt{y}) d y=\sin \sqrt{y}-\sqrt{y} \cos \sqrt{y}+C$ .

Question 61

What is shown in the following graph of $\int_{0}^{1}\left(1-e^{-x}\right) d x$ ?

A)The right approximation with n = 2.
B)The midpoint approximation with n = 2.
C)The trapezoid approximation with n = 2.
D)The left approximation with n = 2.

Question 62

Consider the ellipse pictured below: The perimeter of the ellipse is given by the integral $\int_{0}^{\pi / 2} 8 \sqrt{1-\frac{3}{4} \sin ^{2} \theta} d \theta$ .It turns out that there is no elementary antiderivative for the function $f(\theta)=8 \sqrt{1-\frac{3}{4} \sin ^{2} \theta}$ , and so the integral must be evaluated numerically.A graph of the integrand f( $\theta$ )is shown below. Calculate the right sum that approximates the definite integral with N = 4 equal divisions of the interval.Round to 4 decimal places.

Question 63

The table below shows the velocity v(t)of a falling object at various times (time t measured in seconds, velocity v(t)measured in meters per second). $\begin{array}{cccc}t & 0 & 1 & 2 \\v(t) & 17 & 23 & 28\end{array}$ The distance the object fell in these three seconds lies within which interval?

A)(68, 75.5)
B)(75.5, 78)
C)(78, 83)
D)(83, 86.5)

Question 64

$\int_{0}^{3} \sin ^{39}(x) d x>\pi$ .

Question 65

Suppose the points $x_{0}, x_{1}, \ldots, x_{n}$ are equally spaced and $\alpha$$=$$x_{0}$$<$$x_{1}$$\text { < }$ ... $<$$x_{n}$$=b$ .
Is $\sum_{i=0}^{n-1}\left(\frac{f\left(x_{i}\right)+f\left(x_{i+1}\right)}{2}\right) \Delta x$ the formula (in terms of f and the $x_{i}$ 's)for the midpoint Riemann sum approximation to $\int_{a}^{b} f(x) d x$ ?

Question 66

Below is the graph of . We have the following data:
For , MID(50)= -12.40537 and TRAP(50)= -12.40041.
For , MID(50)= 21.36379 and TRAP(50)= 21.35097.
Using this data alone, what is the best upper bound you can give for ?

Question 67

Using two subdivisions, find the left approximation to .Round to 4 decimal places.

Question 68

You want to estimate $\int_{0}^{1} \cos \left(\theta^{2}\right) d \theta$ by finding values, A and B, such that $A<\int_{0}^{1} \cos$ $\left(\theta^{2}\right) d \theta$ < B , with $B-A$ being as small as possible.Which method should you use to find A?

A)The trapezoid rule.
B)The midpoint rule.
C)The left rule.
D)The right rule.

Question 69

Find the exact value of $\int_{1}^{3} x \sqrt[3]{x-1} d x$ .

A) $\frac{3}{7}(2)^{7 / 3}-\frac{3}{4}(2)^{4 / 3}+\frac{9}{28}$
B) $\frac{3}{7}(2)^{7 / 3}+\frac{3}{4}(2)^{4 / 3}-\frac{33}{28}$
C) $\frac{3}{7}(2)^{7 / 3}+\frac{3}{4}(2)^{4 / 3}$
D) $\frac{3}{7}(2)^{7 / 3}-\frac{3}{4}(2)^{4 / 3}$

Question 70

Consider the definite integral $\int_{0}^{2 / 5} \frac{1}{4+25 x^{2}} d x$ .Compute the integral using the fundamental theorem of calculus and using the midpoint rule with n = 20.How far apart are your answers?

A)Within 0.03 but not within 0.003
B)Within 0.003 but not within 0.0003
C)Within 0.0003 but not within 0.00003
D)Within 0.00003 but not within 0.000003

Question 71

The midpoint rule gives exact answers for linear functions, no matter how many subdivisions are used.

Question 72

For any given function, TRAP(n)is always more accurate than LEFT(n).

Question 73

The table below shows the velocity v(t)of a falling object at various times (time t measured in seconds, velocity v(t)measured in meters per second). $\begin{array}{ccccc}t & 0 & 1 & 2 & 3 \\v(t) & 19 & 25 & 30 & 34\end{array}$ Due to air resistance, the object's acceleration is decreasing.What does this tell you about the shape of the graph of v(t)?

A)It is concave down
B)It is concave up
C)Neither of the above

Question 74

Consider the definite integral $\int_{-\pi / 4}^{\pi / 4} \sin ^{2} \theta d \theta$ .Compute the integral using the fundamental theorem of calculus and using the trapeziod rule with n = 20.How far apart are your answers?

A)Within 0.1 but not within 0.01
B)Within 0.0001 but not within 0.00001
C)Within 0.001 but not within 0.0001
D)Within 0.01 but not within 0.001

Question 75

Suppose the points $x_{0}, x_{1}, \ldots, x_{n}$ are equally spaced and $\alpha$$=$$x_{0}$$<$$x_{1}$$\text { < }$ ... $<$$x_{n}$$=b$ .What is the formula (in terms of f and the $x_{i}$ 's)for the right Riemann sum approximation to $\int_{a}^{b} f(x) d x$ ?

A) $\sum_{i=a}^{b} f\left(x_{i}\right) \Delta x$
B) $\sum_{i=a+n}^{b+n} f\left(x_{i}\right) \Delta x$
C) $\sum_{i=1}^{n} f\left(x_{i}\right) \Delta x$
D) $\sum_{i=0}^{n-1} f\left(x_{i}\right) \Delta x$

Question 76

Consider the function $f(x)=(1-x)^{1 / 3}$ .Does the midpoint approximation give an exact answer, an overestimate, or an underestimate?

A)An overestimate
B)An underestimate
C)An exact answer
D)Cannot tell

Question 77

A drug is being administered intravenously to a patient at a constant rate of 2 mg/hr.The following table shows the rate of change of the amount of the drug in the patient's body at half-hour intervals.Initially there is none of the drug in the patient's body.
Use the trapezoid rule to estimate of the total amount of the drug in the patient's body after four hours.Round to 2 decimal places.

Question 78

Suppose that a computer takes $10^{-7}$ seconds to add two numbers together, and it takes $10^{-5}$ seconds to multiply two numbers together.The computer is asked to integrate the function $f(x)=3 \cdot x^{2}$ from 0 to 1 using left hand sums with n divisions.As a function of n, let T(n)denote the time used by the computer to do the calculation.Compute T(n).(The computer figures x² as x · x.)

A) $2 n \times 10^{-5}+(n-1) \times 10^{-7}$
B) $2 n \times 10^{-7}+2(n-1) \times 10^{-5}$
C) $n \times 10^{-5}+n \times 10^{-7}$
D) $n \times 10^{-5}+2 n \times 10^{-7}$

Question 79

Last Monday we hired a typist to work from 8am to 12 noon.His typing speed decreased between 8am and his 10am cup of coffee, and increased again afterwards, between 10am and noon.His instantaneous speed (measured in characters per second)was measured each hour and the results are given below:
You want to estimate the total number of characters typed between 8am and 12 noon.Find an upper estimate using Reimann sums.

Question 80

Evaluate $\int_{-7}^{18} \arctan (x) d x$ "symbolically" (plug in the limits but don't evaluate).Hint: Integrate by parts using $\frac{d}{d x} \arctan x=\frac{1}{1+x^{2}}$ .

A) $18 \arctan 18-7 \arctan (-7)+\ln \left(1+18^{2}\right)+\ln \left(1+7^{2}\right)$
B) $18 \arctan 18+7 \arctan (-7)-\ln \left(1+18^{2}\right)-\ln \left(1+7^{2}\right)$
C) $18 \arctan 18-7 \arctan (-7)+\frac{1}{2} \ln \left(1+18^{2}\right)-\frac{1}{2} \ln \left(1+7^{2}\right)$
D) $18 \arctan 18+7 \arctan (-7)-\frac{1}{2} \ln \left(1+18^{2}\right)+\frac{1}{2} \ln \left(1+7^{2}\right)$