Question 1

$\text { Find the indefinite integral } \int \frac { x ^ { 2 } } { 4 x ^ { 3 } + 9 } d x \text {. }$

Accepted Answer

A)  $\frac { x ^ { 3 } } { 4 x ^ { 4 } + 9 x } + C$ 
B)  $\ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
C)  $\frac { 1 } { 4 } \ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
D)  $\frac { 1 } { 12 } \ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
E)  $12 \ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
A)  $\frac { x ^ { 3 } } { 4 x ^ { 4 } + 9 x } + C$ 
B)  $\ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
C)  $\frac { 1 } { 4 } \ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
D)  $\frac { 1 } { 12 } \ln \left| 4 x ^ { 3 } + 9 \right| + C$ 
E)  $12 \ln \left| 4 x ^ { 3 } + 9 \right| + C$

Question 2

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { - 1 } ^ { 1 } ( 1 - | u | ) d u$

Accepted Answer

A)  $- 1$ 
B)  1 
C)  2 
D)  4 
E)  15 
A)  $- 1$ 
B)  1 
C)  2 
D)  4 
E)  15

Question 3

graph of f consists of line segments, as shown in the figure. Evaluate the definite  $\text { integral } \int _ { 0 } ^ { 11 } f ( x ) d x \text { using geometric formulas. }$

Accepted Answer

A)  4 
B)  3 
C)  1 
D)  2 
E)  5
= 
A)  4 
B)  3 
C)  1 
D)  2 
E)  5
=

Question 4

$\text { Evaluate the integral } \int _ { \sqrt { 3 } } ^ { 3 } \frac { 12 } { 9 + x ^ { 2 } } d x \text {. }$

Accepted Answer

A)  $\frac { 2 } { 3 \pi }$ 
B)  $\frac { 1 } { 3 } \pi$ 
C)  $\sqrt { \frac { 1 } { 3 } \pi }$ 
D)  $\sqrt { \frac { 1 } { 3 \pi } }$ 
E)  $\frac { 2 } { 3 } \pi$ 
A)  $\frac { 2 } { 3 \pi }$ 
B)  $\frac { 1 } { 3 } \pi$ 
C)  $\sqrt { \frac { 1 } { 3 } \pi }$ 
D)  $\sqrt { \frac { 1 } { 3 \pi } }$ 
E)  $\frac { 2 } { 3 } \pi$

Question 5

The sales S (in thousands of units) of a seasonal product are given by the model $S = 73.24 + 44.29 \sin \frac { \pi t } { 6 }$ where $t$ is the time in months, with $t = 1$ corresponding to January. Find the average sales for the first quarter $( 0 \leq t \leq 3 )$. Round your answer to three decimal places.

Accepted Answer

A)  $90.916$ thousand units 
B)  $92.084$ thousand units 
C)  $101.436$ thousand units 
D)  $129.632$ thousand units 
E)  $72.486$ thousand units 
A)  $90.916$ thousand units 
B)  $92.084$ thousand units 
C)  $101.436$ thousand units 
D)  $129.632$ thousand units 
E)  $72.486$ thousand units

Question 6

$\text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. }$

Accepted Answer

A)  $\frac { 1 } { 8 } \ln ( 5 )$ 
B)  $\frac { 8 } { \ln ( 5 ) }$ 
C)  $\frac { - 1 } { 5 } \ln ( 8 )$ 
D)  $8 \ln ( 5 )$ 
E)  $\frac { 1 } { 5 } \ln ( 8 )$ 
A)  $\frac { 1 } { 8 } \ln ( 5 )$ 
B)  $\frac { 8 } { \ln ( 5 ) }$ 
C)  $\frac { - 1 } { 5 } \ln ( 8 )$ 
D)  $8 \ln ( 5 )$ 
E)  $\frac { 1 } { 5 } \ln ( 8 )$

Question 7

Use the properties of summation and Theorem 5.2 to evaluate the sum. $$\sum _ { i = 1 } ^ { 23 } ( 2 i + 5 )$$

Accepted Answer

A)  667 
B)  552 
C)  575 
D)  1311 
E)  391 
A)  667 
B)  552 
C)  575 
D)  1311 
E)  391

Question 8

$\text { The force } F \text { (in newtons) of a hydraulic cylinder in a press is proportional to the }$ square of $\sec x$ where $x$ is the distance (in meters) that the cylinder is extended in its cycle. The domain of $F$ is $[ 0 , \pi / 3 ]$ and $F ( 0 ) = 400$. Find the average force exerted by the press over the interval $[ 0 , \pi / 3 ]$. Round your answer to the nearest whole number.

Accepted Answer

A)  221 newtons 
B)  110 newtons 
C)  540 newtons 
D)  180 newtons 
E)  662 newtons 
A)  221 newtons 
B)  110 newtons 
C)  540 newtons 
D)  180 newtons 
E)  662 newtons

Question 9

Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function $4 x ^ { 2 } - x - 1 \text { and the } x \text {-axis over the interval } ( 4,14 ) \text {. Round }$ your answer to the nearest integer.

Accepted Answer

A)  2925 
B)  3325 
C)  3000 
D)  3250 
E)  3125 
A)  2925 
B)  3325 
C)  3000 
D)  3250 
E)  3125

Question 10

$\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x$

Accepted Answer

A) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$ 
B) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \ln \left| x ^ { 12 } \right| + C$ 
C)  $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$ 
D) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| x ^ { 12 } \right| + C$ 
E) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| x ^ { 12 } \right| + C$ 
A) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$ 
B) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \ln \left| x ^ { 12 } \right| + C$ 
C)  $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$ 
D) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| x ^ { 12 } \right| + C$ 
E) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| x ^ { 12 } \right| + C$

Question 11

$\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x$

Accepted Answer

A)  $\frac { ( \ln x ) ^ { 7 } } { 6 } + C$ 
B)  $\frac { ( \ln x ) ^ { 7 } } { 7 } + C$ 
C)  $\frac { ( \ln x ) ^ { 6 } } { 6 } + C$ 
D)  $6 ( \ln x ) ^ { 5 } + C$ 
E)  $\frac { 7 \ln x } { x } + C$ 
A)  $\frac { ( \ln x ) ^ { 7 } } { 6 } + C$ 
B)  $\frac { ( \ln x ) ^ { 7 } } { 7 } + C$ 
C)  $\frac { ( \ln x ) ^ { 6 } } { 6 } + C$ 
D)  $6 ( \ln x ) ^ { 5 } + C$ 
E)  $\frac { 7 \ln x } { x } + C$

Question 12

Evaluate the following definite integral by the limit definition. $$\int _ { - 2 } ^ { 4 } \left( 9 s ^ { 2 } + 4 \right) d s$$

Accepted Answer

A)  $- 216$ 
B)  240 
C)  $- 240$ 
D)  192 
E)  216 
A)  $- 216$ 
B)  240 
C)  $- 240$ 
D)  192 
E)  216

Question 13

Find table lists several measurements gathered in an experiment to approximate an unknown continuous function $y = f ( x )$. Approximate the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ using the Trapezoidal Rule. Round your answer to three decimal places.
$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.75 & 2.00 \
\hline y & 4.12 & 4.27 & 4.62 & 6.37 & 6.82 & 7.47 & 7.72 & 8.67 & 8.92 \
\hline
\end{array}$

Accepted Answer

A)  $13.115$ 
B)  $12.000$ 
C)  $7.373$ 
D)  $104.920$ 
E)  $24.600$ 
A)  $13.115$ 
B)  $12.000$ 
C)  $7.373$ 
D)  $104.920$ 
E)  $24.600$

Question 14

The maker of an automobile advertises that it takes $13 \text { seconds to accelerate from } 25$ kilometers per hour to 75 kilometers per hour. Assuming constant acceleration, compute the distance, in meters, the car travels during the 13 seconds. Round your answer to two decimal places.

Accepted Answer

A)  $27.69 \mathrm {~m}$ 
B)  $361.11 \mathrm {~m}$ 
C)  $180.56 \mathrm {~m}$ 
D)  $234.72 \mathrm {~m}$ 
E)  $90.28 \mathrm {~m}$ 
A)  $27.69 \mathrm {~m}$ 
B)  $361.11 \mathrm {~m}$ 
C)  $180.56 \mathrm {~m}$ 
D)  $234.72 \mathrm {~m}$ 
E)  $90.28 \mathrm {~m}$

Question 15

Evaluate the definite integral of the algebraic function. 
$\int _ { 7 } ^ { 10 } \left( s ^ { \frac { 7 } { 4 } } + s ^ { \frac { 4 } { 7 } } \right) d s$ Use a graphing utility to verify your results.

Accepted Answer

A) $\frac { 400 ( 10 ) ^ { \frac { 3 } { 4 } } - 196 ( 7 ) ^ { \frac { 3 } { 4 } } + 70 ( 10 ) ^ { \frac { 4 } { 7 } } - 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 28 }$ 
B) $\frac { 400 ( 10 ) ^ { \frac { 3 } { 4 } } - 196 ( 7 ) ^ { \frac { 3 } { 4 } } + 70 ( 10 ) ^ { \frac { 4 } { 7 } } - 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 11 }$ 
C) $\frac { 400 ( 10 ) ^ { \frac { 3 } { 4 } } + 196 ( 7 ) ^ { \frac { 3 } { 4 } } + 70 ( 10 ) ^ { \frac { 4 } { 7 } } + 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 11 }$ 
D) $\frac { - 400 ( 10 ) ^ { \frac { 3 } { 4 } } + 196 ( 7 ) ^ { \frac { 3 } { 4 } } - 70 ( 10 ) ^ { \frac { 4 } { 7 } } + 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 11 }$ 
E) $\frac { - 400 ( 10 ) ^ { \frac { 3 } { 4 } } + 196 ( 7 ) ^ { \frac { 3 } { 4 } } - 70 ( 10 ) ^ { \frac { 4 } { 7 } } + 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 28 }$ 
A) $\frac { 400 ( 10 ) ^ { \frac { 3 } { 4 } } - 196 ( 7 ) ^ { \frac { 3 } { 4 } } + 70 ( 10 ) ^ { \frac { 4 } { 7 } } - 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 28 }$ 
B) $\frac { 400 ( 10 ) ^ { \frac { 3 } { 4 } } - 196 ( 7 ) ^ { \frac { 3 } { 4 } } + 70 ( 10 ) ^ { \frac { 4 } { 7 } } - 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 11 }$ 
C) $\frac { 400 ( 10 ) ^ { \frac { 3 } { 4 } } + 196 ( 7 ) ^ { \frac { 3 } { 4 } } + 70 ( 10 ) ^ { \frac { 4 } { 7 } } + 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 11 }$ 
D) $\frac { - 400 ( 10 ) ^ { \frac { 3 } { 4 } } + 196 ( 7 ) ^ { \frac { 3 } { 4 } } - 70 ( 10 ) ^ { \frac { 4 } { 7 } } + 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 11 }$ 
E) $\frac { - 400 ( 10 ) ^ { \frac { 3 } { 4 } } + 196 ( 7 ) ^ { \frac { 3 } { 4 } } - 70 ( 10 ) ^ { \frac { 4 } { 7 } } + 49 ( 7 ) ^ { \frac { 4 } { 7 } } } { 28 }$

Question 16

$\text { Find the time required for an object to cool from } 340 ^ { \circ } \mathrm { F } \text { to } 250 ^ { \circ } \mathrm { F } \text { by evaluating }$ $t = \frac { 10 } { \ln 2 } \int _ { 250 } ^ { 340 } \frac { 1 } { T - 70 } d T$, where $t$ is time in minutes. Round your answer to four decimal places.

Accepted Answer

A)  $6.4346$ minutes 
B)  $4.4251$ minutes 
C)  $3.6907$ minutes 
D)  $4.0598$ minutes 
E)  $5.8496$ minutes 
A)  $6.4346$ minutes 
B)  $4.4251$ minutes 
C)  $3.6907$ minutes 
D)  $4.0598$ minutes 
E)  $5.8496$ minutes

Question 17

Find the area of the region bounded by the graphs of the equations $y = x ^ { 3 } + x , x = 4 , y = 0$. Round your answer to the nearest whole number.

Accepted Answer

A)  768 
B)  96 
C)  49 
D)  72 
E)  16 
A)  768 
B)  96 
C)  49 
D)  72 
E)  16

Question 18

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 20 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.
$\int _ { 1 } ^ { 2 } \left( 4 z ^ { 2 } + 7 \right) d z$

Accepted Answer

A)  The Trapezoidal rule gives $16.3400$ and Simpson's rule gives 17,1518 . 
B)  The Trapezoidal rule gives $16.3400$ and Simpson's rule gives 16.3333. 
C)  The Trapezoidal rule gives $- 16.3350$ and Simpson's rule gives $- 16.3333$. 
D)  The Trapezoidal rule gives $16.3350$ and Simpson's rule gives 16.3333. 
E)  The Trapezoidal rule gives $16.3350$ and Simpson's rule gives 17.1518. 
A)  The Trapezoidal rule gives $16.3400$ and Simpson's rule gives 17,1518 . 
B)  The Trapezoidal rule gives $16.3400$ and Simpson's rule gives 16.3333. 
C)  The Trapezoidal rule gives $- 16.3350$ and Simpson's rule gives $- 16.3333$. 
D)  The Trapezoidal rule gives $16.3350$ and Simpson's rule gives 16.3333. 
E)  The Trapezoidal rule gives $16.3350$ and Simpson's rule gives 17.1518.

Question 19

Find the indefinite integral. $$\int \cos 4 x e ^ { \sin 4 x } d x$$

Accepted Answer

A)  $e ^ { \sin 4 x } + C$ 
B)  $4 \sin 4 x \cdot \cos 4 x e ^ { 4 x } + c$ 
C)  $\frac { 1 } { 4 } e ^ { \sin 4 x } + C$ 
D)  $\frac { 1 } { 4 } e ^ { - \cos 4 x } + C$ 
E)  $4 \sin 4 x \cdot e ^ { - \cos 4 x } + C$ 
A)  $e ^ { \sin 4 x } + C$ 
B)  $4 \sin 4 x \cdot \cos 4 x e ^ { 4 x } + c$ 
C)  $\frac { 1 } { 4 } e ^ { \sin 4 x } + C$ 
D)  $\frac { 1 } { 4 } e ^ { - \cos 4 x } + C$ 
E)  $4 \sin 4 x \cdot e ^ { - \cos 4 x } + C$

Question 20

Find the indefinite integral. $$\int \frac { 1 } { \sqrt { 9 - 16 x ^ { 2 } } } d x$$

Accepted Answer

A)  $\frac { 1 } { 4 } \arcsin \left( \frac { 4 x } { 3 } \right) + C$ 
B)  $\frac { 1 } { 4 } \arcsin ( 9 x ) + C$ 
C)  $4 \arcsin ( 9 x ) + C$ 
D)  $4 \arcsin \left( \frac { 4 x } { 3 } \right) + C$ 
E)  $\frac { 1 } { 4 } \arcsin ( 16 x ) + C$ 
A)  $\frac { 1 } { 4 } \arcsin \left( \frac { 4 x } { 3 } \right) + C$ 
B)  $\frac { 1 } { 4 } \arcsin ( 9 x ) + C$ 
C)  $4 \arcsin ( 9 x ) + C$ 
D)  $4 \arcsin \left( \frac { 4 x } { 3 } \right) + C$ 
E)  $\frac { 1 } { 4 } \arcsin ( 16 x ) + C$

$\text { Find the indefinite integral } \int \frac { x ^ { 2 } } { 4 x ^ { 3 } + 9 } d x \text {. }$

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { - 1 } ^ { 1 } ( 1 - | u | ) d u$

$\text { Evaluate the integral } \int _ { \sqrt { 3 } } ^ { 3 } \frac { 12 } { 9 + x ^ { 2 } } d x \text {. }$

$\text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. }$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 23 } ( 2 i + 5 )$

Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function $4 x ^ { 2 } - x - 1 \text { and the } x \text {-axis over the interval } ( 4,14 ) \text {. Round }$ your answer to the nearest integer.

$\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x$

$\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x$

Evaluate the following definite integral by the limit definition. $\int _ { - 2 } ^ { 4 } \left( 9 s ^ { 2 } + 4 \right) d s$

The maker of an automobile advertises that it takes $13 \text { seconds to accelerate from } 25$ kilometers per hour to 75 kilometers per hour. Assuming constant acceleration, compute the distance, in meters, the car travels during the 13 seconds. Round your answer to two decimal places.

Evaluate the definite integral of the algebraic function. $\int _ { 7 } ^ { 10 } \left( s ^ { \frac { 7 } { 4 } } + s ^ { \frac { 4 } { 7 } } \right) d s$ Use a graphing utility to verify your results.

Find the area of the region bounded by the graphs of the equations $y = x ^ { 3 } + x , x = 4 , y = 0$ . Round your answer to the nearest whole number.

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 20 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. $\int _ { 1 } ^ { 2 } \left( 4 z ^ { 2 } + 7 \right) d z$

Find the indefinite integral. $\int \cos 4 x e ^ { \sin 4 x } d x$

Find the indefinite integral. $\int \frac { 1 } { \sqrt { 9 - 16 x ^ { 2 } } } d x$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 5: Antiderivatives and Indefinite Integration

Find the indefinite integral ∫x24x3+9dx. \text { Find the indefinite integral } \int \frac { x ^ { 2 } } { 4 x ^ { 3 } + 9 } d x \text {. } Find the indefinite integral ∫4x3+9x2​dx.

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. ∫−11(1−∣u∣)du\int _ { - 1 } ^ { 1 } ( 1 - | u | ) d u∫−11​(1−∣u∣)du

graph of f consists of line segments, as shown in the figure. Evaluate the definite integral ∫011f(x)dx using geometric formulas. \text { integral } \int _ { 0 } ^ { 11 } f ( x ) d x \text { using geometric formulas. } integral ∫011​f(x)dx using geometric formulas.

Evaluate the integral ∫33129+x2dx. \text { Evaluate the integral } \int _ { \sqrt { 3 } } ^ { 3 } \frac { 12 } { 9 + x ^ { 2 } } d x \text {. } Evaluate the integral ∫3​3​9+x212​dx.

Evaluate the definite integral ∫ee51xln⁡(x)8dx. \text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. } Evaluate the definite integral ∫ee5​xln(x)81​dx.

Use the properties of summation and Theorem 5.2 to evaluate the sum. ∑i=123(2i+5)\sum _ { i = 1 } ^ { 23 } ( 2 i + 5 )∑i=123​(2i+5)

Find ∫1xln⁡(x12)dx\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x Find ∫xln(x12)1​dx

Find the indefinite integral ∫(ln⁡x)6xdx\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x Find the indefinite integral ∫x(lnx)6​dx

Evaluate the following definite integral by the limit definition. ∫−24(9s2+4)ds\int _ { - 2 } ^ { 4 } \left( 9 s ^ { 2 } + 4 \right) d s∫−24​(9s2+4)ds

Evaluate the definite integral of the algebraic function. ∫710(s74+s47)ds\int _ { 7 } ^ { 10 } \left( s ^ { \frac { 7 } { 4 } } + s ^ { \frac { 4 } { 7 } } \right) d s∫710​(s47​+s74​)ds Use a graphing utility to verify your results.

Find the area of the region bounded by the graphs of the equations y=x3+x,x=4,y=0y = x ^ { 3 } + x , x = 4 , y = 0y=x3+x,x=4,y=0 . Round your answer to the nearest whole number.

Find the indefinite integral. ∫cos⁡4xesin⁡4xdx\int \cos 4 x e ^ { \sin 4 x } d x∫cos4xesin4xdx

Find the indefinite integral. ∫19−16x2dx\int \frac { 1 } { \sqrt { 9 - 16 x ^ { 2 } } } d x∫9−16x2​1​dx

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

$\text { Find the indefinite integral } \int \frac { x ^ { 2 } } { 4 x ^ { 3 } + 9 } d x \text {. }$

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { - 1 } ^ { 1 } ( 1 - | u | ) d u$

$\text { Evaluate the integral } \int _ { \sqrt { 3 } } ^ { 3 } \frac { 12 } { 9 + x ^ { 2 } } d x \text {. }$

$\text { Evaluate the definite integral } \int _ { e } ^ { e ^ { 5 } } \frac { 1 } { x \ln ( x ) ^ { 8 } } d x \text {. }$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 23 } ( 2 i + 5 )$

$\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x$

$\text { Find the indefinite integral } \int \frac { ( \ln x ) ^ { 6 } } { x } d x$

Evaluate the following definite integral by the limit definition. $\int _ { - 2 } ^ { 4 } \left( 9 s ^ { 2 } + 4 \right) d s$

Evaluate the definite integral of the algebraic function. $\int _ { 7 } ^ { 10 } \left( s ^ { \frac { 7 } { 4 } } + s ^ { \frac { 4 } { 7 } } \right) d s$ Use a graphing utility to verify your results.

Find the area of the region bounded by the graphs of the equations $y = x ^ { 3 } + x , x = 4 , y = 0$ . Round your answer to the nearest whole number.

Find the indefinite integral. $\int \cos 4 x e ^ { \sin 4 x } d x$

Find the indefinite integral. $\int \frac { 1 } { \sqrt { 9 - 16 x ^ { 2 } } } d x$