Question 1

The solution to the differential equation $\frac { d y } { d x } = x + 2 \sin x$ satisfying the boundary condition $y = 4$ when $x = 0$ is

Accepted Answer

A)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x - 4$ 
B)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x - 2$ 
C)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x + 2$ 
D)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x + 4$ 
E)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x + 6$ 
A)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x - 4$ 
B)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x - 2$ 
C)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x + 2$ 
D)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x + 4$ 
E)  $y = \frac { x ^ { 2 } } { 2 } - 2 \cos x + 6$

Question 2

In a regular partition of the interval [2, 8] into 10 subintervals, then the length of each subintervals Δx is
​

Accepted Answer

A) 1/5 
B) 3/5 
C) 4/5 
D) 1 
E) 6/5 
A) 1/5 
B) 3/5 
C) 4/5 
D) 1 
E) 6/5

Question 3

The solution to the differential equation $\frac { d y } { d x } = - 3 y$ satisfying the boundary condition y = 3 when x = 0 is

Accepted Answer

A)  $y = e ^ { 3 x } + 3$ 
B)  $y = e ^ { - 3 x } + 3$ 
C)  $y = 3 e ^ { 3 x }$ 
D)  $y = - 3 e ^ { - 3 x }$ 
E)  $y = 3 e ^ { - 3 x }$ 
A)  $y = e ^ { 3 x } + 3$ 
B)  $y = e ^ { - 3 x } + 3$ 
C)  $y = 3 e ^ { 3 x }$ 
D)  $y = - 3 e ^ { - 3 x }$ 
E)  $y = 3 e ^ { - 3 x }$

Question 4

Let A denote the area enclosed by the graph $f ( x ) = ( x - 1 ) ^ { 2 },$ the x-axis, and the lines x = 2 and x = 9. Graphing the region and using plane geometry, we can find that A is

Accepted Answer

A)  
 $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } - 1 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$ 
B)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } + 1 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$ 
C)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } + 2 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$ 
D)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 9 k } { n } + 2 \right) ^ { 2 } \left( \frac { 9 } { n } \right)$ 
E)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 9 k } { n } + 1 \right) ^ { 2 } \left( \frac { 9 } { n } \right)$ 
A)  
 $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } - 1 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$ 
B)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } + 1 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$ 
C)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } + 2 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$ 
D)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 9 k } { n } + 2 \right) ^ { 2 } \left( \frac { 9 } { n } \right)$ 
E)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 9 k } { n } + 1 \right) ^ { 2 } \left( \frac { 9 } { n } \right)$

Question 5

The f definite integral $\int _ { - 4 } ^ { 4 } \left( \sin x + x ^ { 3 } \right) d x$ is

Accepted Answer

A)  $- 3$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1 
A)  $- 3$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1

Question 6

In a regular partition of the interval [-4, 12] into 8 subintervals, then the length of each subinterval Δx is
​

Accepted Answer

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

Question 7

Suppose S4 is the upper sum of the area enclosed by the graph $( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = 4 by partitioning [1, 4] into three subintervals [1, 2], [2, 3], and [3, 4]. Then S4 is

Accepted Answer

A) 6 
B)  $\frac { 9 } { 8 }$ 
C)  $\frac { 11 } { 6 }$ 
D) 2 
E)  $\frac { 13 } { 6 }$ 
A) 6 
B)  $\frac { 9 } { 8 }$ 
C)  $\frac { 11 } { 6 }$ 
D) 2 
E)  $\frac { 13 } { 6 }$

Question 8

The indefinite integral $\int \sin ^ { 3 } x \cos x d x$ is

Accepted Answer

A)  $\frac { \sin ^ { 4 } x } { 4 } + C$ 
B)  $- \frac { \cos ^ { 4 } x \sin ^ { 2 } x } { 8 } + C$ 
C)  $- \frac { \cos ^ { 3 } x \sin x } { 4 } + C$ 
D)  $- \frac { \cos ^ { 3 } x } { 4 } + C$ 
E)  $\frac { \cos ^ { 4 } x } { 4 } + C$ 
A)  $\frac { \sin ^ { 4 } x } { 4 } + C$ 
B)  $- \frac { \cos ^ { 4 } x \sin ^ { 2 } x } { 8 } + C$ 
C)  $- \frac { \cos ^ { 3 } x \sin x } { 4 } + C$ 
D)  $- \frac { \cos ^ { 3 } x } { 4 } + C$ 
E)  $\frac { \cos ^ { 4 } x } { 4 } + C$

Question 9

The antiderivative $\int 4 \sec ^ { 2 } x d x$ is

Accepted Answer

A)  $\int 4 \sec ^ { 2 } x d x$ 
B)  $\frac { 4 ( \tan x ) ^ { 3 } } { 3 } + C$ 
C)  $2 ( \tan x ) ^ { 2 } + C$ 
D)  $2 ( \sec x ) ^ { 2 } + C$ 
E)  $4 \tan x + C$ 
A)  $\int 4 \sec ^ { 2 } x d x$ 
B)  $\frac { 4 ( \tan x ) ^ { 3 } } { 3 } + C$ 
C)  $2 ( \tan x ) ^ { 2 } + C$ 
D)  $2 ( \sec x ) ^ { 2 } + C$ 
E)  $4 \tan x + C$

Question 10

The antiderivative $\int \frac { x ^ { 2 } - 3 } { 2 x } d x$ is

Accepted Answer

A)  $\frac { x } { 2 } - \frac { 3 } { 2 x } + C$ 
B)  $\frac { x ^ { 2 } } { 2 } - \frac { 3 } { 2 x ^ { 2 } } + C$ 
C)  $\frac { x ^ { 2 } } { 4 } - \frac { 3 } { 2 } \ln | x | + C$ 
D)  $\frac { x ^ { 2 } } { 2 } - \frac { 3 } { 2 } \ln | x | + C$ 
E)  $\frac { x ^ { 2 } } { 2 } - \frac { 3 x } { 2 } + C$ 
A)  $\frac { x } { 2 } - \frac { 3 } { 2 x } + C$ 
B)  $\frac { x ^ { 2 } } { 2 } - \frac { 3 } { 2 x ^ { 2 } } + C$ 
C)  $\frac { x ^ { 2 } } { 4 } - \frac { 3 } { 2 } \ln | x | + C$ 
D)  $\frac { x ^ { 2 } } { 2 } - \frac { 3 } { 2 } \ln | x | + C$ 
E)  $\frac { x ^ { 2 } } { 2 } - \frac { 3 x } { 2 } + C$

Question 11

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { e } \ln x d x$ are

Accepted Answer

A)  $0,1$ 
B)  $0 , e$ 
C)  $1 , e$ 
D)  $\frac { 1 } { \mathrm { e } } , 1$ 
E)  $0 , \frac { 1 } { \mathrm { e } }$ 
A)  $0,1$ 
B)  $0 , e$ 
C)  $1 , e$ 
D)  $\frac { 1 } { \mathrm { e } } , 1$ 
E)  $0 , \frac { 1 } { \mathrm { e } }$

Question 12

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)  $- \frac { 49 } { 11 }$ 
B)  $- \frac { 49 } { 6 }$ 
C)  $\frac { 49 } { 6 }$ 
D)  $\frac { 49 } { 11 }$ 
E) 7 
A)  $- \frac { 49 } { 11 }$ 
B)  $- \frac { 49 } { 6 }$ 
C)  $\frac { 49 } { 6 }$ 
D)  $\frac { 49 } { 11 }$ 
E) 7

Question 13

The antiderivative $\int ( \cos x - \sin x ) d x$ is

Accepted Answer

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

Question 14

The antiderivative $\int \sqrt [ 6 ] { x ^ { 5 } } d x$ is

Accepted Answer

A)  $\frac { 5 x ^ { \frac { 11 } { 5 } } } { 11 } + C$ 
B)  $\frac { x ^ { 30 } } { 30 } + C$ 
C)  $\frac { x ^ { 31 } } { 31 } + C$ 
D)  $\frac { 5 x ^ { \frac { 11 } { 6 } } } { 11 } + C$ 
E)  $\frac { 6 x ^ { \frac { 11 } { 6 } } } { 11 } + C$ 
A)  $\frac { 5 x ^ { \frac { 11 } { 5 } } } { 11 } + C$ 
B)  $\frac { x ^ { 30 } } { 30 } + C$ 
C)  $\frac { x ^ { 31 } } { 31 } + C$ 
D)  $\frac { 5 x ^ { \frac { 11 } { 6 } } } { 11 } + C$ 
E)  $\frac { 6 x ^ { \frac { 11 } { 6 } } } { 11 } + C$

Question 15

Let A denote the area enclosed by the graph $f ( x ) = \sqrt { 1 - x ^ { 2 } },$ the x-axis, and the lines $x = 0$ and $x = 1$ . Graphing the region and using plane geometry, we can find that A is

Accepted Answer

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

Question 16

If $\int _ { - 2 } ^ { 3 } f ( x ) d x = \frac { 3 } { 2 },$ then $\int _ { - 3 } ^ { 2 } 8 f ( x ) d x$ is

Accepted Answer

A) -12 
B) 12 
C)  $16 / 3$ 
D) -16/3 
E) Impossible to determine 
A) -12 
B) 12 
C)  $16 / 3$ 
D) -16/3 
E) Impossible to determine

Question 17

If $\int _ { 2 } ^ { 1 } 5 f ( x ) d x = 5,$ then $\int _ { 1 } ^ { 2 } 2 f ( x ) d x$ is

Accepted Answer

A) -5 
B) -2 
C) 2 
D) 5 
E) Impossible to determine 
A) -5 
B) -2 
C) 2 
D) 5 
E) Impossible to determine

Question 18

Compute this definite integral using geometric methods: $\int _ { - 7 } ^ { - 4 } \sqrt { 9 - ( x + 4 ) ^ { 2 } } d x$ .

Accepted Answer

A)  $9 \pi / 4$ 
B)  $9 \pi / 2$ 
C)  $9 \pi$ 
D)  $- 9 \pi / 2$ 
E)  $- 9 \pi / 4$ 
A)  $9 \pi / 4$ 
B)  $9 \pi / 2$ 
C)  $9 \pi$ 
D)  $- 9 \pi / 2$ 
E)  $- 9 \pi / 4$

Question 19

The indefinite integral $\int 4 \sin x \sqrt [ 3 ] { 1 + \cos x } d x$ is

Accepted Answer

A)  $3 ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
B)  $\frac { 4 } { 3 } ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
C)  $- 3 ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
D)  $\frac { 16 } { 9 } ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
E)  $\frac { 9 } { 16 } ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
A)  $3 ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
B)  $\frac { 4 } { 3 } ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
C)  $- 3 ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
D)  $\frac { 16 } { 9 } ( 1 + \cos x ) ^ { 4 / 3 } + C$ 
E)  $\frac { 9 } { 16 } ( 1 + \cos x ) ^ { 4 / 3 } + C$

Question 20

The value of the definite integral $\int _ { e ^ { - 1 } } ^ { e } e ^ { \pi } d x$ is

Accepted Answer

A)  $e ^ { \pi } - e ^ { - \pi }$ 
B)  $e ^ { \pi e } - e ^ { - \pi \theta ^ { - 1 } }$ 
C)  $e ^ { 1 + \pi } - e ^ { 1 - \pi }$ 
D)  $e ^ { \pi + 1 } - e ^ { \pi - 1 }$ 
E)  $\frac { \mathrm { e } ^ { \pi + 1 } } { \pi + 1 } \left( e - e ^ { - 1 } \right)$ 
A)  $e ^ { \pi } - e ^ { - \pi }$ 
B)  $e ^ { \pi e } - e ^ { - \pi \theta ^ { - 1 } }$ 
C)  $e ^ { 1 + \pi } - e ^ { 1 - \pi }$ 
D)  $e ^ { \pi + 1 } - e ^ { \pi - 1 }$ 
E)  $\frac { \mathrm { e } ^ { \pi + 1 } } { \pi + 1 } \left( e - e ^ { - 1 } \right)$

The solution to the differential equation $\frac { d y } { d x } = x + 2 \sin x$ satisfying the boundary condition $y = 4$ when $x = 0$ is

In a regular partition of the interval [2, 8] into 10 subintervals, then the length of each subintervals Δx is

The solution to the differential equation $\frac { d y } { d x } = - 3 y$ satisfying the boundary condition y = 3 when x = 0 is

Let A denote the area enclosed by the graph $f ( x ) = ( x - 1 ) ^ { 2 },$ the x-axis, and the lines x = 2 and x = 9. Graphing the region and using plane geometry, we can find that A is

The f definite integral $\int _ { - 4 } ^ { 4 } \left( \sin x + x ^ { 3 } \right) d x$ is

In a regular partition of the interval [-4, 12] into 8 subintervals, then the length of each subinterval Δx is

Suppose S₄ is the upper sum of the area enclosed by the graph $( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = 4 by partitioning [1, 4] into three subintervals [1, 2], [2, 3], and [3, 4]. Then S₄ is

The indefinite integral $\int \sin ^ { 3 } x \cos x d x$ is

The antiderivative $\int 4 \sec ^ { 2 } x d x$ is

The antiderivative $\int \frac { x ^ { 2 } - 3 } { 2 x } d x$ is

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { e } \ln x d x$ are

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

The antiderivative $\int ( \cos x - \sin x ) d x$ is

The antiderivative $\int \sqrt [ 6 ] { x ^ { 5 } } d x$ is

Let A denote the area enclosed by the graph $f ( x ) = \sqrt { 1 - x ^ { 2 } },$ the x-axis, and the lines $x = 0$ and $x = 1$ . Graphing the region and using plane geometry, we can find that A is

If $\int _ { - 2 } ^ { 3 } f ( x ) d x = \frac { 3 } { 2 },$ then $\int _ { - 3 } ^ { 2 } 8 f ( x ) d x$ is

If $\int _ { 2 } ^ { 1 } 5 f ( x ) d x = 5,$ then $\int _ { 1 } ^ { 2 } 2 f ( x ) d x$ is

Compute this definite integral using geometric methods: $\int _ { - 7 } ^ { - 4 } \sqrt { 9 - ( x + 4 ) ^ { 2 } } d x$ .

The indefinite integral $\int 4 \sin x \sqrt [ 3 ] { 1 + \cos x } d x$ is

The value of the definite integral $\int _ { e ^ { - 1 } } ^ { e } e ^ { \pi } d x$ is

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Exam 6: The Integral

The solution to the differential equation dydx=x+2sin⁡x\frac { d y } { d x } = x + 2 \sin xdxdy​=x+2sinx satisfying the boundary condition y=4y = 4y=4 when x=0x = 0x=0 is

In a regular partition of the interval [2, 8] into 10 subintervals, then the length of each subintervals Δx is ​

The solution to the differential equation dydx=−3y\frac { d y } { d x } = - 3 ydxdy​=−3y satisfying the boundary condition y = 3 when x = 0 is

Let A denote the area enclosed by the graph f(x)=(x−1)2,f ( x ) = ( x - 1 ) ^ { 2 },f(x)=(x−1)2, the x-axis, and the lines x = 2 and x = 9. Graphing the region and using plane geometry, we can find that A is

The f definite integral ∫−44(sin⁡x+x3)dx\int _ { - 4 } ^ { 4 } \left( \sin x + x ^ { 3 } \right) d x∫−44​(sinx+x3)dx is

In a regular partition of the interval [-4, 12] into 8 subintervals, then the length of each subinterval Δx is ​

Suppose S4 is the upper sum of the area enclosed by the graph (x)=1x( x ) = \frac { 1 } { x }(x)=x1​ , the x-axis, and the lines x = 1 and x = 4 by partitioning [1, 4] into three subintervals [1, 2], [2, 3], and [3, 4]. Then S4 is

The indefinite integral ∫sin⁡3xcos⁡xdx\int \sin ^ { 3 } x \cos x d x∫sin3xcosxdx is

The antiderivative ∫4sec⁡2xdx\int 4 \sec ^ { 2 } x d x∫4sec2xdx is

The antiderivative ∫x2−32xdx\int \frac { x ^ { 2 } - 3 } { 2 x } d x∫2xx2−3​dx is

The bounds m and M used in the Bounds on an Integral Theorem for ∫1eln⁡xdx\int _ { 1 } ^ { e } \ln x d x∫1e​lnxdx are

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

The antiderivative ∫(cos⁡x−sin⁡x)dx\int ( \cos x - \sin x ) d x∫(cosx−sinx)dx is

The antiderivative ∫x56dx\int \sqrt [ 6 ] { x ^ { 5 } } d x∫6x5​dx is

Let A denote the area enclosed by the graph f(x)=1−x2,f ( x ) = \sqrt { 1 - x ^ { 2 } },f(x)=1−x2​, the x-axis, and the lines x=0x = 0x=0 and x=1x = 1x=1 . Graphing the region and using plane geometry, we can find that A is

If ∫−23f(x)dx=32,\int _ { - 2 } ^ { 3 } f ( x ) d x = \frac { 3 } { 2 },∫−23​f(x)dx=23​, then ∫−328f(x)dx\int _ { - 3 } ^ { 2 } 8 f ( x ) d x∫−32​8f(x)dx is

If ∫215f(x)dx=5,\int _ { 2 } ^ { 1 } 5 f ( x ) d x = 5,∫21​5f(x)dx=5, then ∫122f(x)dx\int _ { 1 } ^ { 2 } 2 f ( x ) d x∫12​2f(x)dx is

Compute this definite integral using geometric methods: ∫−7−49−(x+4)2dx\int _ { - 7 } ^ { - 4 } \sqrt { 9 - ( x + 4 ) ^ { 2 } } d x∫−7−4​9−(x+4)2​dx .

The indefinite integral ∫4sin⁡x1+cos⁡x3dx\int 4 \sin x \sqrt [ 3 ] { 1 + \cos x } d x∫4sinx31+cosx​dx is

The value of the definite integral ∫e−1eeπdx\int _ { e ^ { - 1 } } ^ { e } e ^ { \pi } d x∫e−1e​eπdx is

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The solution to the differential equation $\frac { d y } { d x } = x + 2 \sin x$ satisfying the boundary condition $y = 4$ when $x = 0$ is

In a regular partition of the interval [2, 8] into 10 subintervals, then the length of each subintervals Δx is

The solution to the differential equation $\frac { d y } { d x } = - 3 y$ satisfying the boundary condition y = 3 when x = 0 is

Let A denote the area enclosed by the graph $f ( x ) = ( x - 1 ) ^ { 2 },$ the x-axis, and the lines x = 2 and x = 9. Graphing the region and using plane geometry, we can find that A is

The f definite integral $\int _ { - 4 } ^ { 4 } \left( \sin x + x ^ { 3 } \right) d x$ is

In a regular partition of the interval [-4, 12] into 8 subintervals, then the length of each subinterval Δx is

Suppose S₄ is the upper sum of the area enclosed by the graph $( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = 4 by partitioning [1, 4] into three subintervals [1, 2], [2, 3], and [3, 4]. Then S₄ is

The indefinite integral $\int \sin ^ { 3 } x \cos x d x$ is

The antiderivative $\int 4 \sec ^ { 2 } x d x$ is

The antiderivative $\int \frac { x ^ { 2 } - 3 } { 2 x } d x$ is

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { e } \ln x d x$ are

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

The antiderivative $\int ( \cos x - \sin x ) d x$ is

The antiderivative $\int \sqrt [ 6 ] { x ^ { 5 } } d x$ is

Let A denote the area enclosed by the graph $f ( x ) = \sqrt { 1 - x ^ { 2 } },$ the x-axis, and the lines $x = 0$ and $x = 1$ . Graphing the region and using plane geometry, we can find that A is

If $\int _ { - 2 } ^ { 3 } f ( x ) d x = \frac { 3 } { 2 },$ then $\int _ { - 3 } ^ { 2 } 8 f ( x ) d x$ is

If $\int _ { 2 } ^ { 1 } 5 f ( x ) d x = 5,$ then $\int _ { 1 } ^ { 2 } 2 f ( x ) d x$ is

Compute this definite integral using geometric methods: $\int _ { - 7 } ^ { - 4 } \sqrt { 9 - ( x + 4 ) ^ { 2 } } d x$ .

The indefinite integral $\int 4 \sin x \sqrt [ 3 ] { 1 + \cos x } d x$ is

The value of the definite integral $\int _ { e ^ { - 1 } } ^ { e } e ^ { \pi } d x$ is