Question 1

Let A denote the area enclosed by the graph $f ( x ) = 4 - | x |,$ the x-axis, and the lines x = -2 and x = 0. Graphing the region and using plane geometry, we can find that A is

Accepted Answer

A) 2 
B) 3 
C) 4 
D) 5 
E) 6 
A) 2 
B) 3 
C) 4 
D) 5 
E) 6

Question 2

Let ƒ be an integrable function on [a,b]. Which of the following is not always true?

Accepted Answer

A)  $\int _ { b } ^ { b } f ( x ) d x = 0$ 
B) ƒ is a continuous function on [a,b]. 
C)  $\int _ { a } ^ { b } f ( x ) d x$ is a real number. 
D)  $\left| \int _ { a } ^ { b } f ( x ) d x \right| \geq 0$ 
E)  $\int _ { a } ^ { b } f ( x ) d x = - \int _ { b } ^ { a } f ( x ) d x$ 
A)  $\int _ { b } ^ { b } f ( x ) d x = 0$ 
B) ƒ is a continuous function on [a,b]. 
C)  $\int _ { a } ^ { b } f ( x ) d x$ is a real number. 
D)  $\left| \int _ { a } ^ { b } f ( x ) d x \right| \geq 0$ 
E)  $\int _ { a } ^ { b } f ( x ) d x = - \int _ { b } ^ { a } f ( x ) d x$

Question 3

Suppose S6 is the lower sum of the area enclosed by the graph $f ( x ) = 10 - x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S6 is

Accepted Answer

A) 18.625 
B) 18.635 
C) 18.645 
D) 18.655 
E) 18.665 
A) 18.625 
B) 18.635 
C) 18.645 
D) 18.655 
E) 18.665

Question 4

Suppose the instantaneous velocity $V ( t )$ of a moving particle is $V ( t ) = t ^ { 2 } - 6 t + 3$ meters per second. Which expression below represents the net displacement of the particle between 2 seconds and 5 seconds?

Accepted Answer

A)  $\int _ { 2 } ^ { 5 } | 2 t - 6 | d t$ 
B)  $\int _ { 2 } ^ { 5 } \left( t ^ { 2 } - 6 t + 3 \right) d t$ 
C)  $V ( 5 ) - V ( 2 )$ 
D)  $\int _ { 2 } ^ { 5 } ( 2 t - 6 ) d t$ 
E)  $\int _ { 2 } ^ { 5 } \left| t ^ { 2 } - 6 t + 3 \right| d t$ 
A)  $\int _ { 2 } ^ { 5 } | 2 t - 6 | d t$ 
B)  $\int _ { 2 } ^ { 5 } \left( t ^ { 2 } - 6 t + 3 \right) d t$ 
C)  $V ( 5 ) - V ( 2 )$ 
D)  $\int _ { 2 } ^ { 5 } ( 2 t - 6 ) d t$ 
E)  $\int _ { 2 } ^ { 5 } \left| t ^ { 2 } - 6 t + 3 \right| d t$

Question 5

If $\int _ { - 1 } ^ { 1 } f ( x ) d x = 6,$ then $\int _ { 0 } ^ { 1 } 2 f ( x ) d x$ is

Accepted Answer

A) -12 
B) -6 
C) 6 
D) 12 
E) Impossible to determine 
A) -12 
B) -6 
C) 6 
D) 12 
E) Impossible to determine

Question 6

Let A denote the area enclosed by the graph $f ( x ) = 10 - x,$ the x-axis, and the lines $x = 3$ and $x = 5$ . Graphing the region and using plane geometry, we can find that A is

Accepted Answer

A) 8 
B) 10 
C) 11 
D) 12 
E) 14 
A) 8 
B) 10 
C) 11 
D) 12 
E) 14

Question 7

A sufficient condition for a function ƒ on [a,b] to be integrable is
​

Accepted Answer

A) ƒ is non-negative. 
B) ƒ is positive. 
C) ƒ is bounded above. 
D) ƒ is bounded below. 
E) ƒ is continuous. 
A) ƒ is non-negative. 
B) ƒ is positive. 
C) ƒ is bounded above. 
D) ƒ is bounded below. 
E) ƒ is continuous.

Question 8

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

Accepted Answer

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

Question 9

Let $\int _ { - 2 } ^ { 4 } f ( x ) d x = 12$ and $\int _ { - 2 } ^ { 0 } f ( x ) d x = 7$ Then $\int _ { 0 } ^ { 4 } f ( x ) d x$ is

Accepted Answer

A) -7 
B) -5 
C) 5 
D) 7 
E) 19 
A) -7 
B) -5 
C) 5 
D) 7 
E) 19

Question 10

Let $\int _ { 1 } ^ { 5 } f ( x ) d x = 4$ and $\int _ { 1 } ^ { 2 } f ( x ) d x = 2$ Then $\int _ { 2 } ^ { 5 } f ( x ) d x$ is

Accepted Answer

A) -2 
B) 2 
C) 4 
D) 6 
E) 8 
A) -2 
B) 2 
C) 4 
D) 6 
E) 8

Question 11

The indefinite integral $\int \frac { d x } { \sqrt { 3 x + 7 } }$ is

Accepted Answer

A)  $\frac { 3 } { 2 } \sqrt { 3 x + 7 } + C$ 
B)  $\frac { 3 } { 2 ( 3 x + 7 ) } + C$ 
C)  $\frac { 2 } { 3 } \sqrt { 3 x + 7 } + C$ 
D)  $\frac { 2 } { 3 ( 3 x + 7 ) ^ { 3 / 2 } } + C$ 
E)  $\frac { 2 } { 3 } ( 3 x + 7 ) ^ { 3 / 2 } + C$ 
A)  $\frac { 3 } { 2 } \sqrt { 3 x + 7 } + C$ 
B)  $\frac { 3 } { 2 ( 3 x + 7 ) } + C$ 
C)  $\frac { 2 } { 3 } \sqrt { 3 x + 7 } + C$ 
D)  $\frac { 2 } { 3 ( 3 x + 7 ) ^ { 3 / 2 } } + C$ 
E)  $\frac { 2 } { 3 } ( 3 x + 7 ) ^ { 3 / 2 } + C$

Question 12

The derivative $\frac { d } { d x } \left[ \int _ { 2 } ^ { x } \frac { \sqrt { t } } { t ^ { 2 } + 1 } d t \right]$ is

Accepted Answer

A)  $\frac { x ^ { 2 } + 1 } { \sqrt { x } }$ 
B)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
C)  $\frac { \sqrt { t } } { t ^ { 2 } + 1 }$ 
D)  $\frac { t ^ { 2 } + 1 } { \sqrt { t } }$ 
E)  $\frac { 2 } { 9 \sqrt { x } }$ 
A)  $\frac { x ^ { 2 } + 1 } { \sqrt { x } }$ 
B)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
C)  $\frac { \sqrt { t } } { t ^ { 2 } + 1 }$ 
D)  $\frac { t ^ { 2 } + 1 } { \sqrt { t } }$ 
E)  $\frac { 2 } { 9 \sqrt { x } }$

Question 13

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$ is

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\pi$ 
E)  $\frac { 7 \pi } { 6 }$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\pi$ 
E)  $\frac { 7 \pi } { 6 }$

Question 14

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6, $ is

Accepted Answer

A) -60 
B) -38 
C) 38 
D) 49 
E) 60 
A) -60 
B) -38 
C) 38 
D) 49 
E) 60

Question 15

If f is an even function, $\int _ { - 5 } ^ { 0 } f ( x ) d x = 6,$ and $\int _ { 0 } ^ { 2 } f ( x ) d x = - 2,$ then $\int _ { 2 } ^ { 5 } f ( x ) d x$ ?

Accepted Answer

A)  $- 2$ 
B) 2 
C) 4 
D) 6 
E) 8 
A)  $- 2$ 
B) 2 
C) 4 
D) 6 
E) 8

Question 16

The indefinite integral $\int \frac { \sin x } { \cos x } d x$ is

Accepted Answer

Question 17

The derivative $\frac { d } { d x } \left[ \int _ { 1 } ^ { x } \ln t d t \right]$ is

Accepted Answer

A)  $\frac { 1 } { x }$ 
B)  $\frac { 1 } { \ln x }$ 
C)  $\ln x$ 
D)  $\ln t$ 
E)  $\frac { 1 } { t }$ 
A)  $\frac { 1 } { x }$ 
B)  $\frac { 1 } { \ln x }$ 
C)  $\ln x$ 
D)  $\ln t$ 
E)  $\frac { 1 } { t }$

Question 18

Suppose S5 is the lower sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S5 is

Accepted Answer

A) 240 
B) 260 
C) 280 
D) 300 
E) 320 
A) 240 
B) 260 
C) 280 
D) 300 
E) 320

Question 19

Let $f ( x ) = \frac { 3 } { x ^ { 2 } } - 1$ If $c \in ( 1,3 )$ such that $\frac { \int _ { 1 } ^ { 3 } \frac { 3 } { x ^ { 2 } } - 1 d x } { 2 } = f ( c ),$ then c is

Accepted Answer

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

Question 20

Suppose S5 is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S5 is

Accepted Answer

A) 260 
B) 340 
C) 420 
D) 480 
E) 500 
A) 260 
B) 340 
C) 420 
D) 480 
E) 500

Let A denote the area enclosed by the graph $f ( x ) = 4 - | x |,$ the x-axis, and the lines x = -2 and x = 0. Graphing the region and using plane geometry, we can find that A is

Let ƒ be an integrable function on [a,b]. Which of the following is not always true?

Suppose S₆ is the lower sum of the area enclosed by the graph $f ( x ) = 10 - x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S₆ is

Suppose the instantaneous velocity $V ( t )$ of a moving particle is $V ( t ) = t ^ { 2 } - 6 t + 3$ meters per second. Which expression below represents the net displacement of the particle between 2 seconds and 5 seconds?

If $\int _ { - 1 } ^ { 1 } f ( x ) d x = 6,$ then $\int _ { 0 } ^ { 1 } 2 f ( x ) d x$ is

Let A denote the area enclosed by the graph $f ( x ) = 10 - x,$ the x-axis, and the lines $x = 3$ and $x = 5$ . Graphing the region and using plane geometry, we can find that A is

A sufficient condition for a function ƒ on [a,b] to be integrable is

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

Let $\int _ { - 2 } ^ { 4 } f ( x ) d x = 12$ and $\int _ { - 2 } ^ { 0 } f ( x ) d x = 7$ Then $\int _ { 0 } ^ { 4 } f ( x ) d x$ is

Let $\int _ { 1 } ^ { 5 } f ( x ) d x = 4$ and $\int _ { 1 } ^ { 2 } f ( x ) d x = 2$ Then $\int _ { 2 } ^ { 5 } f ( x ) d x$ is

The indefinite integral $\int \frac { d x } { \sqrt { 3 x + 7 } }$ is

The derivative $\frac { d } { d x } \left[ \int _ { 2 } ^ { x } \frac { \sqrt { t } } { t ^ { 2 } + 1 } d t \right]$ is

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$ is

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

If f is an even function, $\int _ { - 5 } ^ { 0 } f ( x ) d x = 6,$ and $\int _ { 0 } ^ { 2 } f ( x ) d x = - 2,$ then $\int _ { 2 } ^ { 5 } f ( x ) d x$ ?

The indefinite integral $\int \frac { \sin x } { \cos x } d x$ is

The derivative $\frac { d } { d x } \left[ \int _ { 1 } ^ { x } \ln t d t \right]$ is

Suppose S₅is the lower sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S₅ is

Let $f ( x ) = \frac { 3 } { x ^ { 2 } } - 1$ If $c \in ( 1,3 )$ such that $\frac { \int _ { 1 } ^ { 3 } \frac { 3 } { x ^ { 2 } } - 1 d x } { 2 } = f ( c ),$ then c is

Suppose S₅is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S₅ is

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Exam 6: The Integral

Let A denote the area enclosed by the graph f(x)=4−∣x∣,f ( x ) = 4 - | x |,f(x)=4−∣x∣, the x-axis, and the lines x = -2 and x = 0. Graphing the region and using plane geometry, we can find that A is

Let ƒ be an integrable function on [a,b]. Which of the following is not always true?

Suppose S6 is the lower sum of the area enclosed by the graph f(x)=10−x2,f ( x ) = 10 - x ^ { 2 },f(x)=10−x2, the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S6 is

Suppose the instantaneous velocity V(t)V ( t )V(t) of a moving particle is V(t)=t2−6t+3V ( t ) = t ^ { 2 } - 6 t + 3V(t)=t2−6t+3 meters per second. Which expression below represents the net displacement of the particle between 2 seconds and 5 seconds?

If ∫−11f(x)dx=6,\int _ { - 1 } ^ { 1 } f ( x ) d x = 6,∫−11​f(x)dx=6, then ∫012f(x)dx\int _ { 0 } ^ { 1 } 2 f ( x ) d x∫01​2f(x)dx is

Let A denote the area enclosed by the graph f(x)=10−x,f ( x ) = 10 - x,f(x)=10−x, the x-axis, and the lines x=3x = 3x=3 and x=5x = 5x=5 . Graphing the region and using plane geometry, we can find that A is

A sufficient condition for a function ƒ on [a,b] to be integrable is ​

Let ∫−11f(x)dx=4\int _ { - 1 } ^ { 1 } f ( x ) d x = 4∫−11​f(x)dx=4 , and ∫14f(x)dx=3,\int _ { 1 } ^ { 4 } f ( x ) d x = 3,∫14​f(x)dx=3, then ∫−11g(x)dx=−6,\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,∫−11​g(x)dx=−6, is

Let ∫−24f(x)dx=12\int _ { - 2 } ^ { 4 } f ( x ) d x = 12∫−24​f(x)dx=12 and ∫−20f(x)dx=7\int _ { - 2 } ^ { 0 } f ( x ) d x = 7∫−20​f(x)dx=7 Then ∫04f(x)dx\int _ { 0 } ^ { 4 } f ( x ) d x∫04​f(x)dx is

Let ∫15f(x)dx=4\int _ { 1 } ^ { 5 } f ( x ) d x = 4∫15​f(x)dx=4 and ∫12f(x)dx=2\int _ { 1 } ^ { 2 } f ( x ) d x = 2∫12​f(x)dx=2 Then ∫25f(x)dx\int _ { 2 } ^ { 5 } f ( x ) d x∫25​f(x)dx is

The indefinite integral ∫dx3x+7\int \frac { d x } { \sqrt { 3 x + 7 } }∫3x+7​dx​ is

The derivative ddx[∫2xtt2+1dt]\frac { d } { d x } \left[ \int _ { 2 } ^ { x } \frac { \sqrt { t } } { t ^ { 2 } + 1 } d t \right]dxd​[∫2x​t2+1t​​dt] is

By part 2 of the Fundamental Theorem of Calculus, ∫01211−x2dx\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x∫021​​1−x2​1​dx is

Let ∫−11f(x)dx=4\int _ { - 1 } ^ { 1 } f ( x ) d x = 4∫−11​f(x)dx=4 , and ∫14f(x)dx=3,\int _ { 1 } ^ { 4 } f ( x ) d x = 3,∫14​f(x)dx=3, then ∫−11g(x)dx=−6,\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6, ∫−11​g(x)dx=−6, is

If f is an even function, ∫−50f(x)dx=6,\int _ { - 5 } ^ { 0 } f ( x ) d x = 6,∫−50​f(x)dx=6, and ∫02f(x)dx=−2,\int _ { 0 } ^ { 2 } f ( x ) d x = - 2,∫02​f(x)dx=−2, then ∫25f(x)dx\int _ { 2 } ^ { 5 } f ( x ) d x∫25​f(x)dx ?

The indefinite integral ∫sin⁡xcos⁡xdx\int \frac { \sin x } { \cos x } d x∫cosxsinx​dx is

The derivative ddx[∫1xln⁡tdt]\frac { d } { d x } \left[ \int _ { 1 } ^ { x } \ln t d t \right]dxd​[∫1x​lntdt] is

Suppose S5 is the lower sum of the area enclosed by the graph f(x)=x2+x,f ( x ) = x ^ { 2 } + x,f(x)=x2+x, the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S5 is

Let f(x)=3x2−1f ( x ) = \frac { 3 } { x ^ { 2 } } - 1f(x)=x23​−1 If c∈(1,3)c \in ( 1,3 )c∈(1,3) such that ∫133x2−1dx2=f(c),\frac { \int _ { 1 } ^ { 3 } \frac { 3 } { x ^ { 2 } } - 1 d x } { 2 } = f ( c ),2∫13​x23​−1dx​=f(c), then c is

Suppose S5 is the upper sum of the area enclosed by the graph f(x)=x2+x,f ( x ) = x ^ { 2 } + x,f(x)=x2+x, the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S5 is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Let A denote the area enclosed by the graph $f ( x ) = 4 - | x |,$ the x-axis, and the lines x = -2 and x = 0. Graphing the region and using plane geometry, we can find that A is

Suppose S₆ is the lower sum of the area enclosed by the graph $f ( x ) = 10 - x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S₆ is

Suppose the instantaneous velocity $V ( t )$ of a moving particle is $V ( t ) = t ^ { 2 } - 6 t + 3$ meters per second. Which expression below represents the net displacement of the particle between 2 seconds and 5 seconds?

If $\int _ { - 1 } ^ { 1 } f ( x ) d x = 6,$ then $\int _ { 0 } ^ { 1 } 2 f ( x ) d x$ is

Let A denote the area enclosed by the graph $f ( x ) = 10 - x,$ the x-axis, and the lines $x = 3$ and $x = 5$ . Graphing the region and using plane geometry, we can find that A is

A sufficient condition for a function ƒ on [a,b] to be integrable is

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

Let $\int _ { - 2 } ^ { 4 } f ( x ) d x = 12$ and $\int _ { - 2 } ^ { 0 } f ( x ) d x = 7$ Then $\int _ { 0 } ^ { 4 } f ( x ) d x$ is

Let $\int _ { 1 } ^ { 5 } f ( x ) d x = 4$ and $\int _ { 1 } ^ { 2 } f ( x ) d x = 2$ Then $\int _ { 2 } ^ { 5 } f ( x ) d x$ is

The indefinite integral $\int \frac { d x } { \sqrt { 3 x + 7 } }$ is

The derivative $\frac { d } { d x } \left[ \int _ { 2 } ^ { x } \frac { \sqrt { t } } { t ^ { 2 } + 1 } d t \right]$ is

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$ is

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

If f is an even function, $\int _ { - 5 } ^ { 0 } f ( x ) d x = 6,$ and $\int _ { 0 } ^ { 2 } f ( x ) d x = - 2,$ then $\int _ { 2 } ^ { 5 } f ( x ) d x$ ?

The indefinite integral $\int \frac { \sin x } { \cos x } d x$ is

The derivative $\frac { d } { d x } \left[ \int _ { 1 } ^ { x } \ln t d t \right]$ is

Suppose S₅is the lower sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S₅ is

Let $f ( x ) = \frac { 3 } { x ^ { 2 } } - 1$ If $c \in ( 1,3 )$ such that $\frac { \int _ { 1 } ^ { 3 } \frac { 3 } { x ^ { 2 } } - 1 d x } { 2 } = f ( c ),$ then c is

Suppose S₅is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S₅ is