Question 1

Find the indefinite integral.
$$\int 3 x \left( 5 x ^ { 2 } + 3 \right) ^ { 6 } d x$$

Accepted Answer

$$\frac { 3 } { 70 } \left( 5 x ^ { 2 } + 3 \right) ^ { 7 } + C$$

Question 2

The marginal cost of manufacturing $x$ yards of a certain fabric is
$$C ^ { \prime } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$$
in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Accepted Answer

A)  $\$ 188,000.00$ 
B)  $\$ 198,000.00$ 
C)  $\$ 178,000.00$ 
D)  $\$ 218,000.00$ 
E)  $\$ 208,000.00$ 
A)  $\$ 188,000.00$ 
B)  $\$ 198,000.00$ 
C)  $\$ 178,000.00$ 
D)  $\$ 218,000.00$ 
E)  $\$ 208,000.00$ B

Question 3

Use the Midpoint Rule with $n = 10$ to approximate the integral.
$$\int _ { 1 } ^ { 2 } \sqrt { 4 + t ^ { 2 } } d t$$

Accepted Answer

$$2.510716$$

Question 4

Evaluate the definite integral.
$$\int _ { - 1 } ^ { 1 } \frac { \sin x } { 6 + x ^ { 2 } } d x$$

Accepted Answer

The answer of Evaluate the definite integral.
\[\int _ { -...

Question 5

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$.
$f ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }$ is the right endpoint

Accepted Answer

The answer of Use the definition of area to find...

Question 6

Find the integral.
$$\int e ^ { 2 x } \sin 5 x d x$$

Accepted Answer

A)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 5 \cos 5 x ) + C$ 
B)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 5 \cos 2 x ) - C$ 
C)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 2 \cos 5 x ) + C$ 
D)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 2 \cos 5 x ) - C$ 
A)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 5 \cos 5 x ) + C$ 
B)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 5 \cos 2 x ) - C$ 
C)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 2 \cos 5 x ) + C$ 
D)  $\frac { 1 } { 29 } e ^ { 2 x } ( 2 \sin 5 x - 2 \cos 5 x ) - C$

Question 7

Evaluate the integral $\int _ { 0 } ^ { \pi / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$.

Accepted Answer

A)  $5 ( e - 1 )$ 
B)  $20 \left( e ^ { 5 } - 1 \right)$ 
C)  $\frac { e - 1 } { 5 }$ 
D)  $\frac { e ^ { 5 } - 1 } { 20 }$ 
A)  $5 ( e - 1 )$ 
B)  $20 \left( e ^ { 5 } - 1 \right)$ 
C)  $\frac { e - 1 } { 5 }$ 
D)  $\frac { e ^ { 5 } - 1 } { 20 }$

Question 8

Evaluate the integral.
$$\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$$

Accepted Answer

A)  $26.4$ 
B)  $31.4$ 
C)  $46.4$ 
D)  $36.4$ 
E)  $41.4$ 
A)  $26.4$ 
B)  $31.4$ 
C)  $46.4$ 
D)  $36.4$ 
E)  $41.4$

Question 9

Find the integral using the indicated substitution.
$$\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x$$

Accepted Answer

The answer of Find the integral using the indicated substitution.
\[\int...

Question 10

Evaluate the integral by making the given substitution.
$$\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$$

Accepted Answer

The answer of Evaluate the integral by making the given...

Question 11

The acceleration function (in $\mathrm { m } / \mathrm { s } ^ { 2 }$ ) and the initial velocity are given for a particle moving along a line. Find the velocity at time $t$ and the distance traveled during the given time interval.
$$a ( t ) = t + 4 , v ( 0 ) = 6,0 \leq t \leq 10$$

Accepted Answer

The answer of The acceleration function (in \(\mathrm { m...

Question 12

Find the indefinite integral.
$$\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$$

Accepted Answer

A)  $\cos ^ { 2 } x + C$ 
B)  $\tan x + C$ 
C)  $\csc ^ { 2 } x + C$ 
D)  $- \cot x + C$ 
A)  $\cos ^ { 2 } x + C$ 
B)  $\tan x + C$ 
C)  $\csc ^ { 2 } x + C$ 
D)  $- \cot x + C$

Question 13

Find the derivative of the function.
$$g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$$

Accepted Answer

The answer of Find the derivative of the function.
\[g (...

Question 14

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Accepted Answer

The answer of Approximate the area under the curve \(y...

Question 15

Evaluate the integral if it exists.
$$\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$$

Accepted Answer

A)  $1 - 10 \ln x + C$ 
B)  $x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D)  $x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these 
A)  $1 - 10 \ln x + C$ 
B)  $x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D)  $x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these

Question 16

Evaluate the integral.
$$\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 0 }...

Question 17

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline w & 0 & 1 & 2 & 3 & 4 & 5 & 6 \
\hline f(w) & 9.7 & 9.1 & 7.7 & 6.1 & 4.2 & -6.6 & -10.3 \
\hline
\end{array}$

Accepted Answer

The answer of The table gives the values of a...

Question 18

Find the integral.
$$\int x \sin 5 x d x$$

Accepted Answer

The answer of Find the integral.
\[\int x \sin 5 x...

Question 19

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after $t$ weeks is
$$\frac { d x } { d t } = 5700 \left( 1 - \frac { 130 } { ( t + 18 ) ^ { 2 } } \right)$$
calculators per week. Production approaches 5,700 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques. Find the number of calculators produced from the beginning of the third week to the end of the fourth week.
Round the answer to the nearest integer.

Accepted Answer

The answer of Alabama Instruments Company has set up a...

Question 20

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$$

Accepted Answer

A)  $\frac { \sqrt { x + 1 } } { x ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 2 }$ 
C) 
$\frac { \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$ 
D)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
E)  none of these 
A)  $\frac { \sqrt { x + 1 } } { x ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 2 }$ 
C) 
$\frac { \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$ 
D)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
E)  none of these

Find the indefinite integral. $\int 3 x \left( 5 x ^ { 2 } + 3 \right) ^ { 6 } d x$

The marginal cost of manufacturing $x$ yards of a certain fabric is $C ^ { \prime } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$ in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Use the Midpoint Rule with $n = 10$ to approximate the integral. $\int _ { 1 } ^ { 2 } \sqrt { 4 + t ^ { 2 } } d t$

Evaluate the definite integral. $\int _ { - 1 } ^ { 1 } \frac { \sin x } { 6 + x ^ { 2 } } d x$

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$ . $f ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }$ is the right endpoint

Find the integral. $\int e ^ { 2 x } \sin 5 x d x$

Evaluate the integral $\int _ { 0 } ^ { \pi / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$ .

Evaluate the integral. $\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$

Find the integral using the indicated substitution. $\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x$

Evaluate the integral by making the given substitution. $\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$

The acceleration function (in $\mathrm { m } / \mathrm { s } ^ { 2 }$ ) and the initial velocity are given for a particle moving along a line. Find the velocity at time $t$ and the distance traveled during the given time interval. $a ( t ) = t + 4 , v ( 0 ) = 6,0 \leq t \leq 10$

Find the indefinite integral. $\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$

Find the derivative of the function. $g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Evaluate the integral. $\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }$

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f(w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

Find the integral. $\int x \sin 5 x d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$

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Exam 4: Applications of Differentiation

Find the indefinite integral. ∫3x(5x2+3)6dx\int 3 x \left( 5 x ^ { 2 } + 3 \right) ^ { 6 } d x∫3x(5x2+3)6dx

The marginal cost of manufacturing xxx yards of a certain fabric is C′(x)=3−0.01x+0.000006x2C ^ { \prime } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }C′(x)=3−0.01x+0.000006x2 in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Use the Midpoint Rule with n=10n = 10n=10 to approximate the integral. ∫124+t2dt\int _ { 1 } ^ { 2 } \sqrt { 4 + t ^ { 2 } } d t∫12​4+t2​dt

Evaluate the definite integral. ∫−11sin⁡x6+x2dx\int _ { - 1 } ^ { 1 } \frac { \sin x } { 6 + x ^ { 2 } } d x∫−11​6+x2sinx​dx

Use the definition of area to find the area of the region under the graph of fff on [a,b][ a , b ][a,b] using the indicated choice of ckc _ { k }ck​ . f(x)=x2+5x+1,[−1,1],ckf ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }f(x)=x2+5x+1,[−1,1],ck​ is the right endpoint

Find the integral. ∫e2xsin⁡5xdx\int e ^ { 2 x } \sin 5 x d x∫e2xsin5xdx

Evaluate the integral ∫0π/20etan⁡5xcos⁡25xdx\int _ { 0 } ^ { \pi / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x∫0π/20​cos25xetan5x​dx .

Evaluate the integral. ∫14x2+7xdx\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x∫14​x​x2+7​dx

Find the integral using the indicated substitution. ∫tan⁡5xsec⁡2xdx,u=tan⁡x\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x∫tan5xsec2xdx,u=tanx

Evaluate the integral by making the given substitution. ∫x2x3+2dx,u=x3+2\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2∫x2x3+2​dx,u=x3+2

Find the indefinite integral. ∫cot⁡2xcos⁡2xdx\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x∫cos2xcot2x​dx

Find the derivative of the function. g(x)=∫cos⁡x9xcos⁡(t3)dtg ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d tg(x)=∫cosx9x​cos(t3)dt

Approximate the area under the curve y=sin⁡xy = \sin xy=sinx from 0 to π2\frac { \pi } { 2 }2π​ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Evaluate the integral if it exists. ∫(5−xx)2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​)2dx

Evaluate the integral. ∫0x/610+cos⁡2θcos⁡2θ\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }∫0x/6​cos2θ10+cos2θ​

The table gives the values of a function obtained from an experiment. Use the values to estimate ∫06f(w)dw\int _ { 0 } ^ { 6 } f ( w ) d w∫06​f(w)dw using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f(w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

Find the integral. ∫xsin⁡5xdx\int x \sin 5 x d x∫xsin5xdx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1xz2z4+1dzh ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d zh(x)=∫1x​​z4+1z2​dz

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find the indefinite integral. $\int 3 x \left( 5 x ^ { 2 } + 3 \right) ^ { 6 } d x$

The marginal cost of manufacturing $x$ yards of a certain fabric is $C ^ { \prime } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$ in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Use the Midpoint Rule with $n = 10$ to approximate the integral. $\int _ { 1 } ^ { 2 } \sqrt { 4 + t ^ { 2 } } d t$

Evaluate the definite integral. $\int _ { - 1 } ^ { 1 } \frac { \sin x } { 6 + x ^ { 2 } } d x$

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$ . $f ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }$ is the right endpoint

Find the integral. $\int e ^ { 2 x } \sin 5 x d x$

Evaluate the integral $\int _ { 0 } ^ { \pi / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$ .

Evaluate the integral. $\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$

Find the integral using the indicated substitution. $\int \tan ^ { 5 } x \sec ^ { 2 } x d x , \quad u = \tan x$

Evaluate the integral by making the given substitution. $\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$

Find the indefinite integral. $\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$

Find the derivative of the function. $g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Evaluate the integral. $\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }$

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints. w 0 1 2 3 4 5 6 f(w) 9.7 9.1 7.7 6.1 4.2 -6.6 -10.3

Find the integral. $\int x \sin 5 x d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$