Question 1

Find the value of the integral $\int _ { 0 } ^ { 1 / 2 } \frac { \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } d x$

Accepted Answer

A)  $\frac { \pi ^ { 2 } } { 36 }$ 
B)  $\frac { \pi } { 36 }$ 
C)  $\frac { 1 } { 36 }$ 
D)  $\frac { \pi } { 6 }$ 
E) 
$\frac { \pi ^ { 2 } } { 72 }$ 
F) $\frac { \pi } { 72 }$

G) $\frac { 1 } { 72 }$

H) $\frac { \pi } { 2 }$ 
A)  $\frac { \pi ^ { 2 } } { 36 }$ 
B)  $\frac { \pi } { 36 }$ 
C)  $\frac { 1 } { 36 }$ 
D)  $\frac { \pi } { 6 }$ 
E) 
$\frac { \pi ^ { 2 } } { 72 }$ 
F) $\frac { \pi } { 72 }$

G) $\frac { 1 } { 72 }$

H) $\frac { \pi } { 2 }$

Question 2

Evaluate the following integrals:
(a) $\int \frac { \sqrt { u } + \sqrt [ 3 ] { u } } { u } d u$ (b) $\int _ { - 2 } ^ { 1 } | 2 x + 1 | d x$ (c) $\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { t ^ { 3 } + 1 } \cdot t ^ { 5 } d t$ (d) $\int _ { - 4 } ^ { 4 } \sqrt { 16 - x ^ { 2 } } d x$

Accepted Answer

The answer of Evaluate the following integrals:
(a) \(\int \frac {...

Question 3

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { 1 } { 9 + s ^ { 2 } } d s$

Accepted Answer

A) 0 
B)  $\frac { \pi } { 12 }$ 
C) $\frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 6 }$ 
E)  $\frac { \pi } { 4 }$ 
F) $\frac { \pi } { 3 }$ 
G) $\pi$

H)1 
A) 0 
B)  $\frac { \pi } { 12 }$ 
C) $\frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 6 }$ 
E)  $\frac { \pi } { 4 }$ 
F) $\frac { \pi } { 3 }$ 
G) $\pi$

H)1

Question 4

Find the value of the integral $\int _ { 0 } ^ { 1 } \frac { e ^ { x } } { e ^ { x } + 1 } d x$

Accepted Answer

A)  $e + 1$ 
B)  $\ln ( e - 1 )$ 
C)  $\frac { e - 1 } { 2 }$ 
D)  $\ln \frac { e + 1 } { 2 }$ 
E)  $\frac { 1 } { 2 } \ln ( e - 1 )$
F) $\frac { e + 1 } { 2 }$ 
 
G) $\frac { 1 } { 2 } \ln ( e + 1 )$

H) $e - 1$ 
A)  $e + 1$ 
B)  $\ln ( e - 1 )$ 
C)  $\frac { e - 1 } { 2 }$ 
D)  $\ln \frac { e + 1 } { 2 }$ 
E)  $\frac { 1 } { 2 } \ln ( e - 1 )$
F) $\frac { e + 1 } { 2 }$ 
 
G) $\frac { 1 } { 2 } \ln ( e + 1 )$

H) $e - 1$

Question 5

Evaluate the following integrals:
(a) $\int _ { 0 } ^ { x / 2 } \sin x \cos ^ { 2 } x d x$ (b) $\int _ { 0 } ^ { x / 2 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$ (c) $\int _ { - 4 } ^ { 4 } \frac { \sin x \cdot \sqrt { 16 - x ^ { 2 } } } { 4 + x ^ { 2 } } d x$ (d) $\int _ { - 2 } ^ { 2 } \left( x ^ { 3 } + 2 \sqrt { 4 - x ^ { 2 } } \right) d x$

Accepted Answer

(a) @#IMG-DLM& (b) 2

Question 6

If $p ( x )$ is measured in grams per centimeter and x is measured in centimeters, what does $\int _ { a } ^ { b } p ( x ) d x$ represent?

Accepted Answer

(a) It represents the distance

Question 7

Evaluate the following integrals:
(a) $\int \left( 2 + \frac { 1 } { \sqrt { t } } \right) \left( 2 - \frac { 1 } { \sqrt { t } } \right) d t$ (b) $\int _ { 0 } ^ { 1 / 2 } \frac { 1 } { \sqrt { 1 - t ^ { 2 } } } d t$ (c) $\int _ { 0 } ^ { 3 } | 2 - x | d x$ (d) $\int _ { \ln 2 } ^ { \operatorname { m3 } } e ^ { 2 x } d x$

Accepted Answer

The answer of Evaluate the following integrals:
(a) \(\int \left( 2...

Question 8

Use Definition 2 to show that the area under the curve $y = 3 - \frac { x } { 2 } , 0 \leq x \leq 2 \text { is } 5$

Accepted Answer

To show that the area under the curve y

Question 9

Find the value of the integral $\int _ { 0 } ^ { 1 } \arcsin t d t$

Accepted Answer

A)  $\pi - 2$ 
B)  $\frac { \pi - 2 } { 4 }$ 
C)  $\frac { \pi - 2 } { 2 }$ 
D) 1 
E)  $2 - \pi$
F) $\frac { 2 - \pi } { 4 }$

G) $\frac { 2 - \pi } { 2 }$ 
H)0 
A)  $\pi - 2$ 
B)  $\frac { \pi - 2 } { 4 }$ 
C)  $\frac { \pi - 2 } { 2 }$ 
D) 1 
E)  $2 - \pi$
F) $\frac { 2 - \pi } { 4 }$

G) $\frac { 2 - \pi } { 2 }$ 
H)0

Question 10

(a) If $f ( x )$ is measured in pounds and x is measured in feet, what are the units of measurement for $\int _ { a } ^ { b } f ( x ) d x ?$ (b) Give a physical interpretation of f (x) and $\int _ { a } ^ { b } f ( x ) d x$

Accepted Answer

(a) ft-lb
(b) f (x)

Question 11

A scientist collects the following data and plots it in the coordinate plane. $\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline x & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 & 5 \
\hline y & 4 & 10 & 8 & 6 & 14 & 10 & 12 \
\hline
\end{array}$
(a) Use Simpson's rule with n = 6 to estimate the area under the graph of a continuous function drawn through these points.(b) If it is known that $- 4 \leq f ^ { ( 4 ) } ( x ) \leq 1$ for all x, estimate the error involved in the approximation in part (a).(c) How large do we have to choose n so that the approximation $S _ { n }$ (Simpson's Rule) to the integral is accurate to within 0.001?

Accepted Answer

The answer of A scientist collects the following data and...

Question 12

Use right endpoints with n = 4 to approximate $\int _ { 1 } ^ { 2 } \ln 2 x d x$ .

Accepted Answer

The answer of Use right endpoints with n = 4...

Question 13

Find the value of the integral $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 1 \right) ^ { 5 } x ^ { 2 } d x$

Accepted Answer

A) 0 
B) 1 
C)  $\frac { 21 } { 2 }$ 
D)  $\frac { 32 } { 9 }$ 
E)  $\frac { 32 } { 3 }$ 
F)2
G) $\frac { 7 } { 2 }$ 
H) $\frac { 3 } { 2 }$ 
A) 0 
B) 1 
C)  $\frac { 21 } { 2 }$ 
D)  $\frac { 32 } { 9 }$ 
E)  $\frac { 32 } { 3 }$ 
F)2
G) $\frac { 7 } { 2 }$ 
H) $\frac { 3 } { 2 }$

Question 14

The graph of a function f is given below. Estimate $\int _ { 0 } ^ { 8 } f ( x ) d x$ using 4 subintervals with sample points:   (a) right endpoints.(b) left endpoints.(c) midpoints.

Accepted Answer

The answer of The graph of a function f is...

Question 15

Express $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ 2 + \left( \frac { 3 i } { n } \right) ^ { 2 } \right] \frac { 3 } { n } \right\}$ as a definite integral.

Accepted Answer

The answer of Express \(\lim _ { n \rightarrow \infty...

Question 16

Evaluate the improper integral $\int _ { 1 } ^ { \infty } x ^ { - 2 } d x$

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B) 1 
C) 2 
D) 4 
E)  $\frac { 1 } { 2 }$
F)3
G) $\frac { 1 } { 3 }$ 
H)Divergent 
A)  $\frac { 1 } { 4 }$ 
B) 1 
C) 2 
D) 4 
E)  $\frac { 1 } { 2 }$
F)3
G) $\frac { 1 } { 3 }$ 
H)Divergent

Question 17

Evaluate $\int \frac { 1 } { x ^ { 2 } - 4 } d x$

Accepted Answer

A)  $\frac { 1 } { 4 } \ln \left| \frac { x + 2 } { x - 2 } \right| + C$ 
B)  $\ln \left| x ^ { 2 } - 4 \right| + C$ 
C)  $4 \ln \left| \frac { x + 2 } { x - 2 } \right| + C$ 
D)  $\frac { 1 } { 4 } \ln \left| \frac { x - 2 } { x + 2 } \right| + C$ 
E)  $\frac { 1 } { 2 } \ln \left| \frac { x + 2 } { x - 2 } \right| + C$ 
F) $\frac { 1 } { 4 } \ln \left| x ^ { 2 } - 4 \right| + C$ 
G) $\frac { 1 } { 2 } \ln \left| \frac { x - 2 } { x + 2 } \right| + C$ 
H) $- \frac { 1 } { x } - 4 x + C$ 
A)  $\frac { 1 } { 4 } \ln \left| \frac { x + 2 } { x - 2 } \right| + C$ 
B)  $\ln \left| x ^ { 2 } - 4 \right| + C$ 
C)  $4 \ln \left| \frac { x + 2 } { x - 2 } \right| + C$ 
D)  $\frac { 1 } { 4 } \ln \left| \frac { x - 2 } { x + 2 } \right| + C$ 
E)  $\frac { 1 } { 2 } \ln \left| \frac { x + 2 } { x - 2 } \right| + C$ 
F) $\frac { 1 } { 4 } \ln \left| x ^ { 2 } - 4 \right| + C$ 
G) $\frac { 1 } { 2 } \ln \left| \frac { x - 2 } { x + 2 } \right| + C$ 
H) $- \frac { 1 } { x } - 4 x + C$

Question 18

Find the value of the integral $\int _ { 0 } ^ { x / 12 } \sin ^ { 2 } u d u$

Accepted Answer

A)  $\frac { \pi - 3 } { 18 }$ 
B)  $\frac { \pi - 3 } { 16 }$ 
C)  $\frac { \pi - 3 } { 24 }$ 
D)  $\frac { \pi - 3 } { 72 }$ 
E)  $\frac { \pi - 3 } { 36 }$ 
F) $\frac { \pi - 3 } { 12 }$ 
G) $\frac { \pi - 3 } { 48 }$ 
H) $\frac { \pi - 3 } { 60 }$ 
A)  $\frac { \pi - 3 } { 18 }$ 
B)  $\frac { \pi - 3 } { 16 }$ 
C)  $\frac { \pi - 3 } { 24 }$ 
D)  $\frac { \pi - 3 } { 72 }$ 
E)  $\frac { \pi - 3 } { 36 }$ 
F) $\frac { \pi - 3 } { 12 }$ 
G) $\frac { \pi - 3 } { 48 }$ 
H) $\frac { \pi - 3 } { 60 }$

Question 19

Evaluate the integral $\int \cos \sqrt { x } d x$

Accepted Answer

A)  $2 \sin \sqrt { x } + C$ 
B)  $2 \sqrt { x } \cos \sqrt { x } + C$ 
C)  $\sqrt { x } ( \cos \sqrt { x } + \sin \sqrt { x } ) + C$ 
D)  $\frac { \cos \sqrt { x } + \sin \sqrt { x } } { \sqrt { x } } + C$ 
E)  $2 ( \sqrt { x } \sin \sqrt { x } + \cos \sqrt { x } ) + C$ 
F) $2 ( \sqrt { x } \cos \sqrt { x } + \sin \sqrt { x } ) + C$

G) $\sqrt { x } \cos \sqrt { x } + \frac { \sin \sqrt { x } } { \sqrt { x } } + C$

H) $\sqrt { x } \sin \sqrt { x } + \frac { \cos \sqrt { x } } { \sqrt { x } } + C$ 
A)  $2 \sin \sqrt { x } + C$ 
B)  $2 \sqrt { x } \cos \sqrt { x } + C$ 
C)  $\sqrt { x } ( \cos \sqrt { x } + \sin \sqrt { x } ) + C$ 
D)  $\frac { \cos \sqrt { x } + \sin \sqrt { x } } { \sqrt { x } } + C$ 
E)  $2 ( \sqrt { x } \sin \sqrt { x } + \cos \sqrt { x } ) + C$ 
F) $2 ( \sqrt { x } \cos \sqrt { x } + \sin \sqrt { x } ) + C$

G) $\sqrt { x } \cos \sqrt { x } + \frac { \sin \sqrt { x } } { \sqrt { x } } + C$

H) $\sqrt { x } \sin \sqrt { x } + \frac { \cos \sqrt { x } } { \sqrt { x } } + C$

Question 20

Use the Midpoint Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

Accepted Answer

A) 16.25 
B) 0.505 
C) 5.4167 
D) 1.5 
E) 5.625
F)11.25
G)5.3033
H)2.5 
A) 16.25 
B) 0.505 
C) 5.4167 
D) 1.5 
E) 5.625
F)11.25
G)5.3033
H)2.5

Find the value of the integral $\int _ { 0 } ^ { 1 / 2 } \frac { \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } d x$

Evaluate the following integrals: (a) $\int \frac { \sqrt { u } + \sqrt [ 3 ] { u } } { u } d u$ (b) $\int _ { - 2 } ^ { 1 } | 2 x + 1 | d x$ (c) $\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { t ^ { 3 } + 1 } \cdot t ^ { 5 } d t$ (d) $\int _ { - 4 } ^ { 4 } \sqrt { 16 - x ^ { 2 } } d x$

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { 1 } { 9 + s ^ { 2 } } d s$

Find the value of the integral $\int _ { 0 } ^ { 1 } \frac { e ^ { x } } { e ^ { x } + 1 } d x$

If $p ( x )$ is measured in grams per centimeter and x is measured in centimeters, what does $\int _ { a } ^ { b } p ( x ) d x$ represent?

Use Definition 2 to show that the area under the curve $y = 3 - \frac { x } { 2 } , 0 \leq x \leq 2 \text { is } 5$

Find the value of the integral $\int _ { 0 } ^ { 1 } \arcsin t d t$

(a) If $f ( x )$ is measured in pounds and x is measured in feet, what are the units of measurement for $\int _ { a } ^ { b } f ( x ) d x ?$ (b) Give a physical interpretation of f (x) and $\int _ { a } ^ { b } f ( x ) d x$

Use right endpoints with n = 4 to approximate $\int _ { 1 } ^ { 2 } \ln 2 x d x$ .

Find the value of the integral $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 1 \right) ^ { 5 } x ^ { 2 } d x$

Express $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ 2 + \left( \frac { 3 i } { n } \right) ^ { 2 } \right] \frac { 3 } { n } \right\}$ as a definite integral.

Evaluate the improper integral $\int _ { 1 } ^ { \infty } x ^ { - 2 } d x$

Evaluate $\int \frac { 1 } { x ^ { 2 } - 4 } d x$

Find the value of the integral $\int _ { 0 } ^ { x / 12 } \sin ^ { 2 } u d u$

Evaluate the integral $\int \cos \sqrt { x } d x$

Use the Midpoint Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

Functions and Models

Limits and Derivatives

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Applications of Differentiation

Applications of Integration

Differential Equations

Infinite Sequences and Series

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Exam 5: Integrals

Find the value of the integral ∫01/2sin⁡−1x1−x2dx\int _ { 0 } ^ { 1 / 2 } \frac { \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } d x∫01/2​1−x2​sin−1x​dx

Find the value of the integral ∫−3319+s2ds\int _ { - 3 } ^ { 3 } \frac { 1 } { 9 + s ^ { 2 } } d s∫−33​9+s21​ds

Find the value of the integral ∫01exex+1dx\int _ { 0 } ^ { 1 } \frac { e ^ { x } } { e ^ { x } + 1 } d x∫01​ex+1ex​dx

If p(x)p ( x )p(x) is measured in grams per centimeter and x is measured in centimeters, what does ∫abp(x)dx\int _ { a } ^ { b } p ( x ) d x∫ab​p(x)dx represent?

Use Definition 2 to show that the area under the curve y=3−x2,0≤x≤2 is 5y = 3 - \frac { x } { 2 } , 0 \leq x \leq 2 \text { is } 5y=3−2x​,0≤x≤2 is 5

Find the value of the integral ∫01arcsin⁡tdt\int _ { 0 } ^ { 1 } \arcsin t d t∫01​arcsintdt

(a) If f(x)f ( x )f(x) is measured in pounds and x is measured in feet, what are the units of measurement for ∫abf(x)dx?\int _ { a } ^ { b } f ( x ) d x ?∫ab​f(x)dx? (b) Give a physical interpretation of f (x) and ∫abf(x)dx\int _ { a } ^ { b } f ( x ) d x∫ab​f(x)dx

Use right endpoints with n = 4 to approximate ∫12ln⁡2xdx\int _ { 1 } ^ { 2 } \ln 2 x d x∫12​ln2xdx .

Find the value of the integral ∫01(x3+1)5x2dx\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 1 \right) ^ { 5 } x ^ { 2 } d x∫01​(x3+1)5x2dx

The graph of a function f is given below. Estimate ∫08f(x)dx\int _ { 0 } ^ { 8 } f ( x ) d x∫08​f(x)dx using 4 subintervals with sample points: (a) right endpoints.(b) left endpoints.(c) midpoints.

Express lim⁡n→∞{∑i=1n[2+(3in)2]3n}\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ 2 + \left( \frac { 3 i } { n } \right) ^ { 2 } \right] \frac { 3 } { n } \right\}limn→∞​{∑i=1n​[2+(n3i​)2]n3​} as a definite integral.

Evaluate the improper integral ∫1∞x−2dx\int _ { 1 } ^ { \infty } x ^ { - 2 } d x∫1∞​x−2dx

Evaluate ∫1x2−4dx\int \frac { 1 } { x ^ { 2 } - 4 } d x∫x2−41​dx

Find the value of the integral ∫0x/12sin⁡2udu\int _ { 0 } ^ { x / 12 } \sin ^ { 2 } u d u∫0x/12​sin2udu

Evaluate the integral ∫cos⁡xdx\int \cos \sqrt { x } d x∫cosx​dx

Use the Midpoint Rule with n = 4 to approximate the integral ∫−222xdx\int _ { - 2 } ^ { 2 } 2 ^ { x } d x∫−22​2xdx

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the value of the integral $\int _ { 0 } ^ { 1 / 2 } \frac { \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } d x$

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { 1 } { 9 + s ^ { 2 } } d s$

Find the value of the integral $\int _ { 0 } ^ { 1 } \frac { e ^ { x } } { e ^ { x } + 1 } d x$

If $p ( x )$ is measured in grams per centimeter and x is measured in centimeters, what does $\int _ { a } ^ { b } p ( x ) d x$ represent?

Use Definition 2 to show that the area under the curve $y = 3 - \frac { x } { 2 } , 0 \leq x \leq 2 \text { is } 5$

Find the value of the integral $\int _ { 0 } ^ { 1 } \arcsin t d t$

(a) If $f ( x )$ is measured in pounds and x is measured in feet, what are the units of measurement for $\int _ { a } ^ { b } f ( x ) d x ?$ (b) Give a physical interpretation of f (x) and $\int _ { a } ^ { b } f ( x ) d x$

Use right endpoints with n = 4 to approximate $\int _ { 1 } ^ { 2 } \ln 2 x d x$ .

Find the value of the integral $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 1 \right) ^ { 5 } x ^ { 2 } d x$

Express $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ 2 + \left( \frac { 3 i } { n } \right) ^ { 2 } \right] \frac { 3 } { n } \right\}$ as a definite integral.

Evaluate the improper integral $\int _ { 1 } ^ { \infty } x ^ { - 2 } d x$

Evaluate $\int \frac { 1 } { x ^ { 2 } - 4 } d x$

Find the value of the integral $\int _ { 0 } ^ { x / 12 } \sin ^ { 2 } u d u$

Evaluate the integral $\int \cos \sqrt { x } d x$

Use the Midpoint Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$