Question 1

The indefinite integral $\int \sec ^ { 2 } ( 6 x ) d x$ is

Accepted Answer

A)  $12 \sec ( 6 x ) \tan ( 6 x ) + C$ 
B)  $\frac { \tan ( 6 x ) } { 6 } + C$ 
C)  $\frac { \sec ^ { 3 } ( 6 x ) } { 18 } + C$ 
D)  $\frac { \sec ^ { 3 } ( 6 x ) } { 6 } + C$ 
E)  $\frac { \tan ^ { 2 } ( 6 x ) } { 6 } + C$ 
A)  $12 \sec ( 6 x ) \tan ( 6 x ) + C$ 
B)  $\frac { \tan ( 6 x ) } { 6 } + C$ 
C)  $\frac { \sec ^ { 3 } ( 6 x ) } { 18 } + C$ 
D)  $\frac { \sec ^ { 3 } ( 6 x ) } { 6 } + C$ 
E)  $\frac { \tan ^ { 2 } ( 6 x ) } { 6 } + C$

Question 2

Suppose S4 is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + 2,$ the x-axis, and the lines x = 0 and x = 2 by partitioning [0, 2] into four subintervals [0, 1/2], [1/2, 1], [1, 3/2], and [3/2, 2]. Then S4 is

Accepted Answer

A) 8.5 
B) 8.25 
C) 8.125 
D) 7.825 
E) 7.75 
A) 8.5 
B) 8.25 
C) 8.125 
D) 7.825 
E) 7.75

Question 3

Let A denote the area enclosed by the graph $( x ) = \frac { 1 } { x + 1 }$ , the x-axis, and the lines x = 1 and x = 5. Graphing the region and using plane geometry, we can find that A is

Accepted Answer

A)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n } \left( \frac { 1 } { \frac { 4 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$ 
B)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { \frac { 4 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$ 
C)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { \frac { 5 k } { n } + 1 } \right) \left( \frac { 5 } { n } \right)$ 
D)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { 4 k } + 2 \right) \left( \frac { 4 } { n } \right)$ 
E)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { \frac { 5 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$ 
A)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n } \left( \frac { 1 } { \frac { 4 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$ 
B)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { \frac { 4 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$ 
C)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { \frac { 5 k } { n } + 1 } \right) \left( \frac { 5 } { n } \right)$ 
D)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { 4 k } + 2 \right) \left( \frac { 4 } { n } \right)$ 
E)  $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { \frac { 5 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$

Question 4

The indefinite integral $\int ( 5 x + 8 ) ^ { 100 }$ is

Accepted Answer

A)  $500 ( 5 x + 8 ) ^ { 99 } + C$ 
B)  $100 ( 5 x + 8 ) ^ { 99 } + C$ 
C)  $\frac { 1 } { 101 } ( 5 x + 8 ) ^ { 101 } + C$ 
D)  $\frac { 5 } { 101 } ( 5 x + 8 ) ^ { 101 } + C$ 
E)  $\frac { 1 } { 505 } ( 5 x + 8 ) ^ { 101 } + C$ 
A)  $500 ( 5 x + 8 ) ^ { 99 } + C$ 
B)  $100 ( 5 x + 8 ) ^ { 99 } + C$ 
C)  $\frac { 1 } { 101 } ( 5 x + 8 ) ^ { 101 } + C$ 
D)  $\frac { 5 } { 101 } ( 5 x + 8 ) ^ { 101 } + C$ 
E)  $\frac { 1 } { 505 } ( 5 x + 8 ) ^ { 101 } + C$

Question 5

The definite integral $\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) ^ { 2 } } d x$ is

Accepted Answer

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

Question 6

The definite integral $\int _ { 0 } ^ { 1 } 2 x ^ { 2 } \sqrt { x ^ { 3 } + 1 } d x$ is

Accepted Answer

A)  $\frac { 2 } { 3 } ( 2 \sqrt { 2 } - 1 )$ 
B)  $\frac { 4 } { 9 } ( 2 \sqrt { 2 } - 1 )$ 
C)  $\frac { 4 } { 3 } ( 2 \sqrt { 2 } - 1 )$ 
D)  $\frac { 3 } { 2 } ( 2 \sqrt { 2 } - 1 )$ 
E)  $\frac { 9 } { 4 } ( 2 \sqrt { 2 } - 1 )$ 
A)  $\frac { 2 } { 3 } ( 2 \sqrt { 2 } - 1 )$ 
B)  $\frac { 4 } { 9 } ( 2 \sqrt { 2 } - 1 )$ 
C)  $\frac { 4 } { 3 } ( 2 \sqrt { 2 } - 1 )$ 
D)  $\frac { 3 } { 2 } ( 2 \sqrt { 2 } - 1 )$ 
E)  $\frac { 9 } { 4 } ( 2 \sqrt { 2 } - 1 )$

Question 7

The value of the definite integral $\int _ { - 2 } ^ { 2 } \ln e d x$ is

Accepted Answer

A) 2 
B)  $\frac { 4 } { e }$ 
C) ln 4 
D)  $\ln 2 - \ln \frac { 1 } { 2 }$ 
E) 4 
A) 2 
B)  $\frac { 4 } { e }$ 
C) ln 4 
D)  $\ln 2 - \ln \frac { 1 } { 2 }$ 
E) 4

Question 8

The antiderivative $\int \left( 1 + \cot ^ { 2 } x \right) d x$ is

Accepted Answer

A)  $x + \frac { \cot ^ { 3 } x } { 3 } + C$ 
B)  $x - \frac { \cot ^ { 3 } x } { 3 } + C$ 
C)  $- \csc ^ { 2 } x + C$ 
D)  $- \cot ^ { 2 } x + C$ 
E)  $- \cot x + C$ 
A)  $x + \frac { \cot ^ { 3 } x } { 3 } + C$ 
B)  $x - \frac { \cot ^ { 3 } x } { 3 } + C$ 
C)  $- \csc ^ { 2 } x + C$ 
D)  $- \cot ^ { 2 } x + C$ 
E)  $- \cot x + C$

Question 9

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { 2 } x ^ { 3 },$ the x-axis, and the lines x = 0 and x = 2. Graphing the region and using plane geometry, we can find that A is

Accepted Answer

A)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$ 
B)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k - 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$ 
C)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 2 } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$ 
D)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$ 
E)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$ 
A)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$ 
B)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k - 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$ 
C)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 2 } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$ 
D)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$ 
E)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$

Question 10

If the half-life of a radioactive isotope is 5000 years, then its decay constant is

Accepted Answer

A)  $- \frac { 2 } { \ln 5000 }$ 
B)  $- \frac { 1 } { \ln 2500 }$ 
C)  $- \frac { \ln 2 } { \ln 5000 }$ 
D)  $- \frac { \ln 2 } { 5000 }$ 
E)  $- \frac { \ln 5000 } { 2 }$ 
A)  $- \frac { 2 } { \ln 5000 }$ 
B)  $- \frac { 1 } { \ln 2500 }$ 
C)  $- \frac { \ln 2 } { \ln 5000 }$ 
D)  $- \frac { \ln 2 } { 5000 }$ 
E)  $- \frac { \ln 5000 } { 2 }$

Question 11

The indefinite integral $\int \sin \left( \frac { x } { 2 } \right) d x$ is

Accepted Answer

A)  $\frac { 1 } { 2 } \cos \left( \frac { x } { 2 } \right) + C$ 
B)  $- \cos \left( \frac { x } { 2 } \right) + C$ 
C)  $- \frac { 1 } { 2 } \cos \left( \frac { x } { 2 } \right) + C$ 
D)  $- 2 \cos \left( \frac { x } { 2 } \right) + C$ 
E)  $2 \cos \left( \frac { x } { 2 } \right) + C$ 
A)  $\frac { 1 } { 2 } \cos \left( \frac { x } { 2 } \right) + C$ 
B)  $- \cos \left( \frac { x } { 2 } \right) + C$ 
C)  $- \frac { 1 } { 2 } \cos \left( \frac { x } { 2 } \right) + C$ 
D)  $- 2 \cos \left( \frac { x } { 2 } \right) + C$ 
E)  $2 \cos \left( \frac { x } { 2 } \right) + C$

Question 12

If $F ( x ) = \int _ { x } ^ { - 3 } ( 1 - 2 t ) d t$ , what is F(0)?

Accepted Answer

A) -12 
B) 12 
C) 9 
D) -6 
E) 6 
A) -12 
B) 12 
C) 9 
D) -6 
E) 6

Question 13

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$ is

Accepted Answer

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

Question 14

The solution to the differential equation $\frac { d y } { d x } = \frac { e ^ { x } } { 2 y }$ satisfying the boundary condition $y = 5$ when $x = 0$ is

Accepted Answer

A)  $y ^ { 2 } = e ^ { x } + 3$ 
B)  $y ^ { 2 } = e ^ { x } + 8$ 
C)  $y ^ { 2 } = e ^ { x } + 15$ 
D)  $y ^ { 2 } = e ^ { x } + 24$ 
E)  $y ^ { 2 } = e ^ { x } + 35$ 
A)  $y ^ { 2 } = e ^ { x } + 3$ 
B)  $y ^ { 2 } = e ^ { x } + 8$ 
C)  $y ^ { 2 } = e ^ { x } + 15$ 
D)  $y ^ { 2 } = e ^ { x } + 24$ 
E)  $y ^ { 2 } = e ^ { x } + 35$

Question 15

The value of the definite integral $\int _ { - \pi } ^ { \pi } \frac { e ^ { 3 } - e ^ { - 3 } } { 2 } d x$ is

Accepted Answer

A) 
 $\left( \frac { e ^ { 4 } } { 8 } - \frac { e ^ { - 2 } } { 4 } \right) \pi$ 
B)  $\left( \frac { e ^ { 4 } } { 4 } - \frac { e ^ { - 2 } } { 2 } \right) \pi$ 
C)  $\left. e ^ { 3 } - e ^ { - 3 } \right) \pi$ 
D)  $\left( \frac { e ^ { 3 } } { 2 } - \frac { e ^ { - 3 } } { 2 } \right) \pi$ 
E)  $2 \pi \left( e ^ { 3 } - e ^ { - 3 } \right)$ 
A) 
 $\left( \frac { e ^ { 4 } } { 8 } - \frac { e ^ { - 2 } } { 4 } \right) \pi$ 
B)  $\left( \frac { e ^ { 4 } } { 4 } - \frac { e ^ { - 2 } } { 2 } \right) \pi$ 
C)  $\left. e ^ { 3 } - e ^ { - 3 } \right) \pi$ 
D)  $\left( \frac { e ^ { 3 } } { 2 } - \frac { e ^ { - 3 } } { 2 } \right) \pi$ 
E)  $2 \pi \left( e ^ { 3 } - e ^ { - 3 } \right)$

Question 16

The value of the definite integral $\int _ { - 4 } ^ { 6 } \frac { 1 } { e } d x$ is

Accepted Answer

A)  $\frac { 2 } { e }$ 
B)  $\frac { 4 } { e }$ 
C)  $\frac { 6 } { e }$ 
D)  $\frac { 8 } { e }$ 
E)  $\frac { 10 } { e }$ 
A)  $\frac { 2 } { e }$ 
B)  $\frac { 4 } { e }$ 
C)  $\frac { 6 } { e }$ 
D)  $\frac { 8 } { e }$ 
E)  $\frac { 10 } { e }$

Question 17

By part 2 of the Fundamental Theorem of Calculus, $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sec x \tan x d x$ is

Accepted Answer

A)  $2 - \sqrt { 2 }$ 
B)  $\sqrt { 2 } - 2$ 
C)  $2 + \sqrt { 2 }$ 
D)  $\frac { \sqrt { 2 } - 2 } { 2 }$ 
E)  $\frac { 2 - \sqrt { 2 } } { 2 }$ 
A)  $2 - \sqrt { 2 }$ 
B)  $\sqrt { 2 } - 2$ 
C)  $2 + \sqrt { 2 }$ 
D)  $\frac { \sqrt { 2 } - 2 } { 2 }$ 
E)  $\frac { 2 - \sqrt { 2 } } { 2 }$

Question 18

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

Accepted Answer

A)  $- \pi ^ { 2 }$ 
B)  $- \pi$ 
C)  $\pi$ 
D)  $\pi ^ { 2 }$ 
E) Impossible to determine 
A)  $- \pi ^ { 2 }$ 
B)  $- \pi$ 
C)  $\pi$ 
D)  $\pi ^ { 2 }$ 
E) Impossible to determine

Question 19

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { \pi } { 2 } } ( 2 x + \cos x ) d x$ is

Accepted Answer

A)  $\frac { \pi ^ { 2 } + 4 } { 4 }$ 
B)  $\frac { \pi ^ { 2 } - 4 } { 4 }$ 
C)  $\frac { \pi ^ { 2 } + 2 } { 4 }$ 
D)  $\frac { \pi ^ { 2 } - 2 } { 4 }$ 
E)  $\pi ^ { 2 }$ 
A)  $\frac { \pi ^ { 2 } + 4 } { 4 }$ 
B)  $\frac { \pi ^ { 2 } - 4 } { 4 }$ 
C)  $\frac { \pi ^ { 2 } + 2 } { 4 }$ 
D)  $\frac { \pi ^ { 2 } - 2 } { 4 }$ 
E)  $\pi ^ { 2 }$

Question 20

Let A denote the area enclosed by the graph $f ( x ) = | x |,$ 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) 1 
B) 1.5 
C) 2 
D) 2.5 
E) 3 
A) 1 
B) 1.5 
C) 2 
D) 2.5 
E) 3

The indefinite integral $\int \sec ^ { 2 } ( 6 x ) d x$ is

Suppose S₄is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + 2,$ the x-axis, and the lines x = 0 and x = 2 by partitioning [0, 2] into four subintervals [0, 1/2], [1/2, 1], [1, 3/2], and [3/2, 2]. Then S₄ is

Let A denote the area enclosed by the graph $( x ) = \frac { 1 } { x + 1 }$ , the x-axis, and the lines x = 1 and x = 5. Graphing the region and using plane geometry, we can find that A is

The indefinite integral $\int ( 5 x + 8 ) ^ { 100 }$ is

The definite integral $\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) ^ { 2 } } d x$ is

The definite integral $\int _ { 0 } ^ { 1 } 2 x ^ { 2 } \sqrt { x ^ { 3 } + 1 } d x$ is

The value of the definite integral $\int _ { - 2 } ^ { 2 } \ln e d x$ is

The antiderivative $\int \left( 1 + \cot ^ { 2 } x \right) d x$ is

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { 2 } x ^ { 3 },$ the x-axis, and the lines x = 0 and x = 2. Graphing the region and using plane geometry, we can find that A is

If the half-life of a radioactive isotope is 5000 years, then its decay constant is

The indefinite integral $\int \sin \left( \frac { x } { 2 } \right) d x$ is

If $F ( x ) = \int _ { x } ^ { - 3 } ( 1 - 2 t ) d t$ , what is F(0)?

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$ is

The solution to the differential equation $\frac { d y } { d x } = \frac { e ^ { x } } { 2 y }$ satisfying the boundary condition $y = 5$ when $x = 0$ is

The value of the definite integral $\int _ { - \pi } ^ { \pi } \frac { e ^ { 3 } - e ^ { - 3 } } { 2 } d x$ is

The value of the definite integral $\int _ { - 4 } ^ { 6 } \frac { 1 } { e } d x$ is

By part 2 of the Fundamental Theorem of Calculus, $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sec x \tan x d x$ is

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

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { \pi } { 2 } } ( 2 x + \cos x ) d x$ is

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

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Exam 6: The Integral

The indefinite integral ∫sec⁡2(6x)dx\int \sec ^ { 2 } ( 6 x ) d x∫sec2(6x)dx is

Suppose S4 is the upper sum of the area enclosed by the graph f(x)=x2+2,f ( x ) = x ^ { 2 } + 2,f(x)=x2+2, the x-axis, and the lines x = 0 and x = 2 by partitioning [0, 2] into four subintervals [0, 1/2], [1/2, 1], [1, 3/2], and [3/2, 2]. Then S4 is

Let A denote the area enclosed by the graph (x)=1x+1( x ) = \frac { 1 } { x + 1 }(x)=x+11​ , the x-axis, and the lines x = 1 and x = 5. Graphing the region and using plane geometry, we can find that A is

The indefinite integral ∫(5x+8)100\int ( 5 x + 8 ) ^ { 100 }∫(5x+8)100 is

The definite integral ∫141x(1+x)2dx\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) ^ { 2 } } d x∫14​x​(1+x​)21​dx is

The definite integral ∫012x2x3+1dx\int _ { 0 } ^ { 1 } 2 x ^ { 2 } \sqrt { x ^ { 3 } + 1 } d x∫01​2x2x3+1​dx is

The value of the definite integral ∫−22ln⁡edx\int _ { - 2 } ^ { 2 } \ln e d x∫−22​lnedx is

The antiderivative ∫(1+cot⁡2x)dx\int \left( 1 + \cot ^ { 2 } x \right) d x∫(1+cot2x)dx is

Let A denote the area enclosed by the graph f(x)=12x3,f ( x ) = \frac { 1 } { 2 } x ^ { 3 },f(x)=21​x3, the x-axis, and the lines x = 0 and x = 2. Graphing the region and using plane geometry, we can find that A is

If the half-life of a radioactive isotope is 5000 years, then its decay constant is

The indefinite integral ∫sin⁡(x2)dx\int \sin \left( \frac { x } { 2 } \right) d x∫sin(2x​)dx is

If F(x)=∫x−3(1−2t)dtF ( x ) = \int _ { x } ^ { - 3 } ( 1 - 2 t ) d tF(x)=∫x−3​(1−2t)dt , what is F(0)?

The derivative ddx[∫x205tdt]\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]dxd​[∫x20​5tdt] is

The solution to the differential equation dydx=ex2y\frac { d y } { d x } = \frac { e ^ { x } } { 2 y }dxdy​=2yex​ satisfying the boundary condition y=5y = 5y=5 when x=0x = 0x=0 is

The value of the definite integral ∫−ππe3−e−32dx\int _ { - \pi } ^ { \pi } \frac { e ^ { 3 } - e ^ { - 3 } } { 2 } d x∫−ππ​2e3−e−3​dx is

The value of the definite integral ∫−461edx\int _ { - 4 } ^ { 6 } \frac { 1 } { e } d x∫−46​e1​dx is

By part 2 of the Fundamental Theorem of Calculus, ∫π4π3sec⁡xtan⁡xdx\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sec x \tan x d x∫4π​3π​​secxtanxdx is

If ∫01f(x)dx=π,\int _ { 0 } ^ { 1 } f ( x ) d x = \pi,∫01​f(x)dx=π, then π2∫10f(x)dx\frac { \pi ^ { 2 } } { \int _ { 1 } ^ { 0 } f ( x ) d x }∫10​f(x)dxπ2​ is

By part 2 of the Fundamental Theorem of Calculus, ∫0π2(2x+cos⁡x)dx\int _ { 0 } ^ { \frac { \pi } { 2 } } ( 2 x + \cos x ) d x∫02π​​(2x+cosx)dx is

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

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

Applications of the Integral

Techniques of Integration

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Parametric Equations; Polar Equations

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Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

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The indefinite integral $\int \sec ^ { 2 } ( 6 x ) d x$ is

Suppose S₄is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + 2,$ the x-axis, and the lines x = 0 and x = 2 by partitioning [0, 2] into four subintervals [0, 1/2], [1/2, 1], [1, 3/2], and [3/2, 2]. Then S₄ is

Let A denote the area enclosed by the graph $( x ) = \frac { 1 } { x + 1 }$ , the x-axis, and the lines x = 1 and x = 5. Graphing the region and using plane geometry, we can find that A is

The indefinite integral $\int ( 5 x + 8 ) ^ { 100 }$ is

The definite integral $\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) ^ { 2 } } d x$ is

The definite integral $\int _ { 0 } ^ { 1 } 2 x ^ { 2 } \sqrt { x ^ { 3 } + 1 } d x$ is

The value of the definite integral $\int _ { - 2 } ^ { 2 } \ln e d x$ is

The antiderivative $\int \left( 1 + \cot ^ { 2 } x \right) d x$ is

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { 2 } x ^ { 3 },$ the x-axis, and the lines x = 0 and x = 2. Graphing the region and using plane geometry, we can find that A is

The indefinite integral $\int \sin \left( \frac { x } { 2 } \right) d x$ is

If $F ( x ) = \int _ { x } ^ { - 3 } ( 1 - 2 t ) d t$ , what is F(0)?

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$ is

The solution to the differential equation $\frac { d y } { d x } = \frac { e ^ { x } } { 2 y }$ satisfying the boundary condition $y = 5$ when $x = 0$ is

The value of the definite integral $\int _ { - \pi } ^ { \pi } \frac { e ^ { 3 } - e ^ { - 3 } } { 2 } d x$ is

The value of the definite integral $\int _ { - 4 } ^ { 6 } \frac { 1 } { e } d x$ is

By part 2 of the Fundamental Theorem of Calculus, $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sec x \tan x d x$ is

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

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { \pi } { 2 } } ( 2 x + \cos x ) d x$ is

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