Question 1

$$\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x$$

Accepted Answer

4

Question 2

Find the limit. $$\lim _ { h \rightarrow 0 } \frac { 1 } { h } \int _ { 2 } ^ { 2 + h } \sqrt { 1 + t ^ { 3 } } d t$$

Accepted Answer

3

Question 3

Find the derivative of the function. $$g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$$

Accepted Answer

$$8 x \sqrt { 1 + x ^ { 4 } }$$

Question 4

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 5

Evaluate the integral. $$\int _ { x / 2 } ^ { x } \cos \left( \frac { 1 } { 2 } x \right) d x$$

Accepted Answer

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

Question 6

Evaluate the integral. $$\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x$$

Accepted Answer

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

Question 7

Evaluate the limit after first finding the sum (as a function of $n$ ) using the summation formulas.
$$\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 4 + \frac { 2 k } { n } \right) ^ { 2 } \left( \frac { 1 } { n } \right)$$

Accepted Answer

The answer of Evaluate the limit after first finding the...

Question 8

Find the indefinite integral.
$$\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x$$

Accepted Answer

A)  $- 2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
B)  $2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
C)  $\left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
A)  $- 2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
B)  $2 \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
C)  $\left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 4 x - 5 \right) ^ { 3 } + C$

Question 9

Evaluate the integral. $$\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$$

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 2 \pi } { 3 }$

Question 10

Find the indefinite integral.
$$\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$$

Accepted Answer

A)  $2 x + \frac { 5 } { 4 x ^ { 4 } } - \frac { 1 } { 4 x ^ { 8 } } + C$ 
B)  $2 x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 6 } + C$ 
C)  $2 x + \frac { 5 } { 4 } x ^ { 4 } - \frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 5 } { 2 x ^ { 2 } } - \frac { 1 } { 3 x ^ { 6 } } + C$ 
A)  $2 x + \frac { 5 } { 4 x ^ { 4 } } - \frac { 1 } { 4 x ^ { 8 } } + C$ 
B)  $2 x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 6 } + C$ 
C)  $2 x + \frac { 5 } { 4 } x ^ { 4 } - \frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 5 } { 2 x ^ { 2 } } - \frac { 1 } { 3 x ^ { 6 } } + C$

Question 11

Find the indefinite integral.
$$\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$$

Accepted Answer

A) 
$2 x + \frac { 5 } { 4 x ^ { 4 } } - \frac { 1 } { 4 x ^ { 8 } } + C$ 
B)  $2 x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 6 } + C$ 
C)  $2 x + \frac { 5 } { 4 } x ^ { 4 } - \frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 5 } { 2 x ^ { 2 } } - \frac { 1 } { 3 x ^ { 6 } } + C$ 
A) 
$2 x + \frac { 5 } { 4 x ^ { 4 } } - \frac { 1 } { 4 x ^ { 8 } } + C$ 
B)  $2 x + \frac { 5 } { 2 } x ^ { 2 } - \frac { 1 } { 3 } x ^ { 6 } + C$ 
C)  $2 x + \frac { 5 } { 4 } x ^ { 4 } - \frac { 1 } { 4 } x ^ { 8 } + C$ 
D)  $2 x + \frac { 5 } { 2 x ^ { 2 } } - \frac { 1 } { 3 x ^ { 6 } } + C$

Question 12

Find the area of the region under the graph of $f$ on $[ a , b ]$. 
$f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$

Accepted Answer

A)  3 
B)  $- 12$ 
C)  12 
D)  $- 3$ 
A)  3 
B)  $- 12$ 
C)  12 
D)  $- 3$

Question 13

Find the derivative of the function. $$F ( x ) = \int _ { x } ^ { x ^ { 2 } } e ^ { t ^ { 5 } } d t$$

Accepted Answer

The answer of Find the derivative of the function. \[F...

Question 14

Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

Accepted Answer

A)  191 
B)  188 
C)  190 
D)  189 
E)  192 
A)  191 
B)  188 
C)  190 
D)  189 
E)  192

Question 15

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 16

Find the indefinite integral. $$\int x \sqrt { x - 8 } d x$$

Accepted Answer

The answer of Find the indefinite integral. \[\int x \sqrt...

Question 17

$$\text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. }$$

Accepted Answer

A sketch of the area between t

Question 18

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

A)  $3.09$ 
B)  $1.09$ 
C)  $4.09$ 
D)  $0.09$ 
E)  $2.09$ 
A)  $3.09$ 
B)  $1.09$ 
C)  $4.09$ 
D)  $0.09$ 
E)  $2.09$

Question 19

Find the indefinite integral. $$\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x$$

Accepted Answer

The answer of Find the indefinite integral. \[\int x ^...

Question 20

Evaluate the Riemann sum for $f ( r ) = 6 - r ^ { 2 } , 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

Accepted Answer

A)  $10.75$ 
B)  $9.75$ 
C)  $8.25$ 
D)  $10.25$ 
E)  $9.25$ 
A)  $10.75$ 
B)  $9.75$ 
C)  $8.25$ 
D)  $10.25$ 
E)  $9.25$

$\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x$

Find the limit. $\lim _ { h \rightarrow 0 } \frac { 1 } { h } \int _ { 2 } ^ { 2 + h } \sqrt { 1 + t ^ { 3 } } d t$

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

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

Evaluate the integral. $\int _ { x / 2 } ^ { x } \cos \left( \frac { 1 } { 2 } x \right) d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x$

Evaluate the limit after first finding the sum (as a function of $n$ ) using the summation formulas. $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 4 + \frac { 2 k } { n } \right) ^ { 2 } \left( \frac { 1 } { n } \right)$

Find the indefinite integral. $\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$

Find the derivative of the function. $F ( x ) = \int _ { x } ^ { x ^ { 2 } } e ^ { t ^ { 5 } } d t$

Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

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

Find the indefinite integral. $\int x \sqrt { x - 8 } d x$

$\text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. }$

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.

Find the indefinite integral. $\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x$

Evaluate the Riemann sum for $f ( r ) = 6 - r ^ { 2 } , 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

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Techniques of Integration

Further Applications of Integration

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Vectors and the Geometry of Space

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Exam 4: Applications of Differentiation

Evaluate ∫−1/21/2x7−2x5+3x3+4x2+x−4x2−1dx\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x Evaluate ∫−1/21/2​x2−1x7−2x5+3x3+4x2+x−4​dx

Find the limit. lim⁡h→01h∫22+h1+t3dt\lim _ { h \rightarrow 0 } \frac { 1 } { h } \int _ { 2 } ^ { 2 + h } \sqrt { 1 + t ^ { 3 } } d tlimh→0​h1​∫22+h​1+t3​dt

Find the derivative of the function. g(x)=∫4x241+t2dtg ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d tg(x)=∫4x2​41+t2​dt

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

Evaluate the integral. ∫x/2xcos⁡(12x)dx\int _ { x / 2 } ^ { x } \cos \left( \frac { 1 } { 2 } x \right) d x∫x/2x​cos(21​x)dx

Evaluate the integral. ∫017x6cos⁡(x7)dx\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x∫01​7x6cos(x7)dx

Find the indefinite integral. ∫(x2+4x−5)2(2x+4)dx\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x∫(x2+4x−5)2(2x+4)dx

Evaluate the integral. ∫01/24dr1−r2\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }∫01/2​1−r2​4dr​

Find the indefinite integral. ∫2x7−5x4+2x7dx\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x∫x72x7−5x4+2​dx

Find the indefinite integral. ∫2x7−5x4+2x7dx\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x∫x72x7−5x4+2​dx

Find the area of the region under the graph of fff on [a,b][ a , b ][a,b] . f(x)=x2−2x+4;[−1,2]f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]f(x)=x2−2x+4;[−1,2]

Find the derivative of the function. F(x)=∫xx2et5dtF ( x ) = \int _ { x } ^ { x ^ { 2 } } e ^ { t ^ { 5 } } d tF(x)=∫xx2​et5dt

Estimate the area from 0 to 5 under the graph of f(x)=49−x2f ( x ) = 49 - x ^ { 2 }f(x)=49−x2 using five approximating rectangles and right endpoints.

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

Find the indefinite integral. ∫xx−8dx\int x \sqrt { x - 8 } d x∫xx−8​dx

Show that ∫0x−5cos⁡2xdx=0 by interpreting the definite integral geometrically. \text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. } Show that ∫0x​−5cos2xdx=0 by interpreting the definite integral geometrically.

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.

Find the indefinite integral. ∫x3sec⁡2x4dx\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x∫x3sec2x4dx

Evaluate the Riemann sum for f(r)=6−r2,0≤r≤2f ( r ) = 6 - r ^ { 2 } , 0 \leq r \leq 2f(r)=6−r2,0≤r≤2 with four subintervals, taking the sample points to be right endpoints.

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

$\text { Evaluate } \int _ { - 1 / 2 } ^ { 1 / 2 } \frac { x ^ { 7 } - 2 x ^ { 5 } + 3 x ^ { 3 } + 4 x ^ { 2 } + x - 4 } { x ^ { 2 } - 1 } d x$

Find the limit. $\lim _ { h \rightarrow 0 } \frac { 1 } { h } \int _ { 2 } ^ { 2 + h } \sqrt { 1 + t ^ { 3 } } d t$

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

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

Evaluate the integral. $\int _ { x / 2 } ^ { x } \cos \left( \frac { 1 } { 2 } x \right) d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } 7 x ^ { 6 } \cos \left( x ^ { 7 } \right) d x$

Find the indefinite integral. $\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } \frac { 4 d r } { \sqrt { 1 - r ^ { 2 } } }$

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Find the indefinite integral. $\int \frac { 2 x ^ { 7 } - 5 x ^ { 4 } + 2 } { x ^ { 7 } } d x$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$

Find the derivative of the function. $F ( x ) = \int _ { x } ^ { x ^ { 2 } } e ^ { t ^ { 5 } } d t$

Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.

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

Find the indefinite integral. $\int x \sqrt { x - 8 } d x$

$\text { Show that } \int _ { 0 } ^ { x } - 5 \cos 2 x d x = 0 \text { by interpreting the definite integral geometrically. }$

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.

Find the indefinite integral. $\int x ^ { 3 } \sec ^ { 2 } x ^ { 4 } d x$

Evaluate the Riemann sum for $f ( r ) = 6 - r ^ { 2 } , 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.