Question 1

Compute this definite integral using geometric methods: $\int _ { 0 } ^ { 4 } \sqrt { 4 - ( x - 2 ) ^ { 2 } } d x$ ?

Accepted Answer

A)  $\pi$ 
B)  $2 \pi$ 
C)  $4 \pi$ 
D)  $6 \pi$ 
E)  $8 \pi$ 
A)  $\pi$ 
B)  $2 \pi$ 
C)  $4 \pi$ 
D)  $6 \pi$ 
E)  $8 \pi$

Question 2

The indefinite integral $\int x \sqrt { x + 1 } d x$ is

Accepted Answer

A)  $\frac { 1 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 3 x - 2 ) + C$ 
B)  $\frac { 2 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 3 x - 2 ) + C$ 
C)  $\frac { 1 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 2 - 3 x ) + C$ 
D)  $\frac { 2 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 3 x - 1 ) + C$ 
E)  $\frac { 1 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 1 - 3 x ) + C$ 
A)  $\frac { 1 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 3 x - 2 ) + C$ 
B)  $\frac { 2 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 3 x - 2 ) + C$ 
C)  $\frac { 1 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 2 - 3 x ) + C$ 
D)  $\frac { 2 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 3 x - 1 ) + C$ 
E)  $\frac { 1 } { 15 } ( x + 1 ) ^ { 3 / 2 } ( 1 - 3 x ) + C$

Question 3

Suppose the instantaneous velocity V(t) of a moving particle is V(t) = t2- 6t + 3 meters per second. Which expression below represents the total distance traveled by 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 4

The antiderivative $\int ( 1 - \sin x ) d x$ is

Accepted Answer

A)  $x - \cos x + C$ 
B)  $x + \cos x + C$ 
C)  $1 - \frac { \sin ^ { 2 } x } { 2 } + C$ 
D)  $x - \frac { \sin ^ { 2 } x } { 2 } + C$ 
E)  $x + \frac { \cos ^ { 2 } x } { 2 } + C$ 
A)  $x - \cos x + C$ 
B)  $x + \cos x + C$ 
C)  $1 - \frac { \sin ^ { 2 } x } { 2 } + C$ 
D)  $x - \frac { \sin ^ { 2 } x } { 2 } + C$ 
E)  $x + \frac { \cos ^ { 2 } x } { 2 } + C$

Question 5

Suppose S4 is the upper sum of the area enclosed by the graph $f ( x ) = 2 x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 4 by partitioning [0, 4] into four subintervals [0, 1], [1, 2], [2, 3], and [3, 4]. Then S4 is

Accepted Answer

A) 26 
B) 28 
C) 32 
D) 48 
E) 60 
A) 26 
B) 28 
C) 32 
D) 48 
E) 60

Question 6

The value of the definite integral $\int _ { 0 } ^ { 2 } \log _ { 2 } 32 d x$ is

Accepted Answer

A) 8 
B) 10 
C) 16 
D) 32 
E) 64 
A) 8 
B) 10 
C) 16 
D) 32 
E) 64

Question 7

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

Accepted Answer

A)  $- \frac { 1 } { 4 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 2 }$ 
E) 0 
A)  $- \frac { 1 } { 4 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 2 }$ 
E) 0

Question 8

Following a massive rainstorm, water flows into a storm water drainage pond for 6 hours. If V(t) denotes the volume of water in the pond t minutes after the start of flow into the pond, which integral represents the net change of water entering the pond between 2 and 3 hours?

Accepted Answer

A)  $\int _ { 2 } ^ { 3 } V ^ { \prime } ( t ) d t$ 
B)  $\int _ { 0 } ^ { 1 } V ^ { \prime } ( t ) d t$ 
C)  $\int _ { 120 } ^ { 180 } V ^ { \prime } ( t ) d t$ 
D)  $\int _ { 2 } ^ { 3 } V ( t ) d t$ 
E)  $\int _ { 120 } ^ { 180 } V ( t ) d t$ 
A)  $\int _ { 2 } ^ { 3 } V ^ { \prime } ( t ) d t$ 
B)  $\int _ { 0 } ^ { 1 } V ^ { \prime } ( t ) d t$ 
C)  $\int _ { 120 } ^ { 180 } V ^ { \prime } ( t ) d t$ 
D)  $\int _ { 2 } ^ { 3 } V ( t ) d t$ 
E)  $\int _ { 120 } ^ { 180 } V ( t ) d t$

Question 9

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

Accepted Answer

A)  $\sin y = - \cos x$ 
B)  $\sin y = - \cos x + 1$ 
C)  $\sin y = - \cos x - 1$ 
D)  $\sin y = - \cos x - 2$ 
E)  $\sin y = - \cos x + 2$ 
A)  $\sin y = - \cos x$ 
B)  $\sin y = - \cos x + 1$ 
C)  $\sin y = - \cos x - 1$ 
D)  $\sin y = - \cos x - 2$ 
E)  $\sin y = - \cos x + 2$

Question 10

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

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4

Question 11

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x$ are

Accepted Answer

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

Question 12

The average value of $f ( x ) = \sec ^ { 2 } x$ on $\left[ 0 , \frac { \pi } { 4 } \right]$ is

Accepted Answer

A)  $\frac { 2 } { \pi }$ 
B)  $\frac { 3 } { \pi }$ 
C)  $\frac { 4 } { \pi }$ 
D)  $\frac { 5 } { \pi }$ 
E)  $\frac { 6 } { \pi }$ 
A)  $\frac { 2 } { \pi }$ 
B)  $\frac { 3 } { \pi }$ 
C)  $\frac { 4 } { \pi }$ 
D)  $\frac { 5 } { \pi }$ 
E)  $\frac { 6 } { \pi }$

Question 13

The indefinite integral $\int \frac { x - 3 } { \sqrt { x ^ { 2 } - 6 x + 5 } } d x$ is

Accepted Answer

A)  $2 \sqrt { x ^ { 2 } - 6 x + 5 } + C$ 
B)  $\sqrt { x ^ { 2 } - 6 x + 5 } + C$ 
C)  $\frac { 3 } { x ^ { 2 } - 6 x + 5 } + C$ 
D)  $2 \left( x ^ { 2 } - 6 x + 5 \right) ^ { 3 / 2 } + C$ 
E)  $\frac { 1 } { 2 } \left( x ^ { 2 } - 6 x + 5 \right) ^ { 1 / 2 } + C$ 
A)  $2 \sqrt { x ^ { 2 } - 6 x + 5 } + C$ 
B)  $\sqrt { x ^ { 2 } - 6 x + 5 } + C$ 
C)  $\frac { 3 } { x ^ { 2 } - 6 x + 5 } + C$ 
D)  $2 \left( x ^ { 2 } - 6 x + 5 \right) ^ { 3 / 2 } + C$ 
E)  $\frac { 1 } { 2 } \left( x ^ { 2 } - 6 x + 5 \right) ^ { 1 / 2 } + C$

Question 14

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

Accepted Answer

A)  $\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )$ 
B)  $\int _ { a } ^ { b } f ( x ) d x = \int _ { a / 2 } ^ { b / 2 } 2 f ( x ) d x$ 
C)  $\int _ { a } ^ { b } f ( x ) d x = \frac { 1 } { 2 } \int _ { 2 a } ^ { 2 b } f ( x ) d x$ 
D)  $\int _ { a } ^ { b } f ( x ) d x > 0$ 
E) is bounded. $\int _ { a } ^ { b } f ( x ) d x$ 
A)  $\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )$ 
B)  $\int _ { a } ^ { b } f ( x ) d x = \int _ { a / 2 } ^ { b / 2 } 2 f ( x ) d x$ 
C)  $\int _ { a } ^ { b } f ( x ) d x = \frac { 1 } { 2 } \int _ { 2 a } ^ { 2 b } f ( x ) d x$ 
D)  $\int _ { a } ^ { b } f ( x ) d x > 0$ 
E) is bounded. $\int _ { a } ^ { b } f ( x ) d x$

Question 15

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

Accepted Answer

A)  $e ^ { 2 } - 1$ 
B)  $e ^ { 2 }$ 
C)  $e + 1$ 
D)  $e$ 
E)  $e - 1$ 
A)  $e ^ { 2 } - 1$ 
B)  $e ^ { 2 }$ 
C)  $e + 1$ 
D)  $e$ 
E)  $e - 1$

Question 16

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

Accepted Answer

A)  $e ^ { x ^ { 2 } } - 1$ 
B)  $e ^ { x ^ { 2 } }$ 
C)  $\frac { e ^ { x ^ { 2 } } } { 2 x }$ 
D)  $2 x e ^ { x ^ { 2 } }$ 
E)  $2 x e ^ { x ^ { 2 } } - 1$ 
A)  $e ^ { x ^ { 2 } } - 1$ 
B)  $e ^ { x ^ { 2 } }$ 
C)  $\frac { e ^ { x ^ { 2 } } } { 2 x }$ 
D)  $2 x e ^ { x ^ { 2 } }$ 
E)  $2 x e ^ { x ^ { 2 } } - 1$

Question 17

If the number of bacteria in a culture grows from 20,000 to 60,000 in 2 hours, then the growth constant is

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $\int _ { a } ^ { a } f ( x ) d x = \int _ { b } ^ { b } f ( x ) d x$ 
B)  $\int _ { a } ^ { b } k d x = k ( b - a )$ 
C)  $\int _ { a } ^ { b } f ( x ) d x$ is bounded. 
D)  $\int _ { a } ^ { b } k d x = - k ( a - b )$ 
E)  $2 \int _ { a } ^ { b } f ( x ) d x > \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$ 
A)  $\int _ { a } ^ { a } f ( x ) d x = \int _ { b } ^ { b } f ( x ) d x$ 
B)  $\int _ { a } ^ { b } k d x = k ( b - a )$ 
C)  $\int _ { a } ^ { b } f ( x ) d x$ is bounded. 
D)  $\int _ { a } ^ { b } k d x = - k ( a - b )$ 
E)  $2 \int _ { a } ^ { b } f ( x ) d x > \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$

Question 19

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

Accepted Answer

A)  $\cos ^ { 2 } x$ 
B)  $\cos ^ { 2 } x - 1$ 
C)  $\sin ^ { 2 } x - 1$ 
D)  $\sin ^ { 2 } x$ 
E)  $\frac { \cos ^ { 3 } x } { 3 } - 1$ 
A)  $\cos ^ { 2 } x$ 
B)  $\cos ^ { 2 } x - 1$ 
C)  $\sin ^ { 2 } x - 1$ 
D)  $\sin ^ { 2 } x$ 
E)  $\frac { \cos ^ { 3 } x } { 3 } - 1$

Question 20

Let ƒ be an integrable function on [a, b]. Which of the following is always true?

Accepted Answer

A)  $\int _ { a } ^ { b } f ( x ) d x > b - a$ 
B) 
 $\int _ { a } ^ { b } f ( x ) d x  \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$ 
D) 
 $\left[ \int _ { a } ^ { b } f ( x ) d x \right] ^ { 2 } > \int _ { a } ^ { b } f ( x ) d x$ 
E)  
 $\int _ { a } ^ { b } f ( x ) d x \leq \left| \int _ { a } ^ { b } f ( x ) d x \right|$ 
A)  $\int _ { a } ^ { b } f ( x ) d x > b - a$ 
B) 
 $\int _ { a } ^ { b } f ( x ) d x  \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$ 
D) 
 $\left[ \int _ { a } ^ { b } f ( x ) d x \right] ^ { 2 } > \int _ { a } ^ { b } f ( x ) d x$ 
E)  
 $\int _ { a } ^ { b } f ( x ) d x \leq \left| \int _ { a } ^ { b } f ( x ) d x \right|$

Compute this definite integral using geometric methods: $\int _ { 0 } ^ { 4 } \sqrt { 4 - ( x - 2 ) ^ { 2 } } d x$ ?

The indefinite integral $\int x \sqrt { x + 1 } d x$ is

Suppose the instantaneous velocity V(t) of a moving particle is V(t) = t²- 6t + 3 meters per second. Which expression below represents the total distance traveled by the particle between 2 seconds and 5 seconds?

The antiderivative $\int ( 1 - \sin x ) d x$ is

Suppose S₄ is the upper sum of the area enclosed by the graph $f ( x ) = 2 x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 4 by partitioning [0, 4] into four subintervals [0, 1], [1, 2], [2, 3], and [3, 4]. Then S₄ is

The value of the definite integral $\int _ { 0 } ^ { 2 } \log _ { 2 } 32 d x$ is

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

Following a massive rainstorm, water flows into a storm water drainage pond for 6 hours. If V(t) denotes the volume of water in the pond t minutes after the start of flow into the pond, which integral represents the net change of water entering the pond between 2 and 3 hours?

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

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

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x$ are

The average value of $f ( x ) = \sec ^ { 2 } x$ on $\left[ 0 , \frac { \pi } { 4 } \right]$ is

The indefinite integral $\int \frac { x - 3 } { \sqrt { x ^ { 2 } - 6 x + 5 } } d x$ is

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

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

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

If the number of bacteria in a culture grows from 20,000 to 60,000 in 2 hours, then the growth constant is

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

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

Let ƒ be an integrable function on [a, b]. Which of the following is always true?

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Exam 6: The Integral

Compute this definite integral using geometric methods: ∫044−(x−2)2dx\int _ { 0 } ^ { 4 } \sqrt { 4 - ( x - 2 ) ^ { 2 } } d x∫04​4−(x−2)2​dx ?

The indefinite integral ∫xx+1dx\int x \sqrt { x + 1 } d x∫xx+1​dx is

Suppose the instantaneous velocity V(t) of a moving particle is V(t) = t2- 6t + 3 meters per second. Which expression below represents the total distance traveled by the particle between 2 seconds and 5 seconds?

The antiderivative ∫(1−sin⁡x)dx\int ( 1 - \sin x ) d x∫(1−sinx)dx is

Suppose S4 is the upper sum of the area enclosed by the graph f(x)=2x2,f ( x ) = 2 x ^ { 2 },f(x)=2x2, the x-axis, and the lines x = 0 and x = 4 by partitioning [0, 4] into four subintervals [0, 1], [1, 2], [2, 3], and [3, 4]. Then S4 is

The value of the definite integral ∫02log⁡232dx\int _ { 0 } ^ { 2 } \log _ { 2 } 32 d x∫02​log2​32dx is

Let f(x)=x3f ( x ) = x ^ { 3 }f(x)=x3 If c∈(−1,1)c \in ( - 1,1 )c∈(−1,1) such that ∫−11x3dx2=f(c),\frac { \int _ { - 1 } ^ { 1 } x ^ { 3 } d x } { 2 } = f ( c ),2∫−11​x3dx​=f(c), then c is

Following a massive rainstorm, water flows into a storm water drainage pond for 6 hours. If V(t) denotes the volume of water in the pond t minutes after the start of flow into the pond, which integral represents the net change of water entering the pond between 2 and 3 hours?

The solution to the differential equation dydx=sin⁡xcos⁡y\frac { d y } { d x } = \frac { \sin x } { \cos y }dxdy​=cosysinx​ satisfying the boundary condition y=π2y = \frac { \pi } { 2 }y=2π​ when x=0x = 0x=0 is

By part 2 of the Fundamental Theorem of Calculus, ∫−22dx\int _ { - 2 } ^ { 2 } d x∫−22​dx is

The bounds m and M used in the Bounds on an Integral Theorem for ∫121x2dx\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x∫12​x21​dx are

The average value of f(x)=sec⁡2xf ( x ) = \sec ^ { 2 } xf(x)=sec2x on [0,π4]\left[ 0 , \frac { \pi } { 4 } \right][0,4π​] is

The indefinite integral ∫x−3x2−6x+5dx\int \frac { x - 3 } { \sqrt { x ^ { 2 } - 6 x + 5 } } d x∫x2−6x+5​x−3​dx is

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

By part 2 of the Fundamental Theorem of Calculus, ∫01exdx\int _ { 0 } ^ { 1 } e ^ { x } d x∫01​exdx is

The derivative ddx[∫0xet2dt]\frac { d } { d x } \left[ \int _ { 0 } ^ { x } e ^ { t ^ { 2 } } d t \right]dxd​[∫0x​et2dt] is

If the number of bacteria in a culture grows from 20,000 to 60,000 in 2 hours, then the growth constant is

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

The derivative ddx[∫0xcos⁡2tdt]\frac { d } { d x } \left[ \int _ { 0 } ^ { x } \cos ^ { 2 } t d t \right]dxd​[∫0x​cos2tdt] is

Let ƒ be an integrable function on [a, b]. Which of the following is always true?

Preparing for Calculus

Limits and Continuity

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Applications of the Integral

Techniques of Integration

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Parametric Equations; Polar Equations

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Vector Functions

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Compute this definite integral using geometric methods: $\int _ { 0 } ^ { 4 } \sqrt { 4 - ( x - 2 ) ^ { 2 } } d x$ ?

The indefinite integral $\int x \sqrt { x + 1 } d x$ is

Suppose the instantaneous velocity V(t) of a moving particle is V(t) = t²- 6t + 3 meters per second. Which expression below represents the total distance traveled by the particle between 2 seconds and 5 seconds?

The antiderivative $\int ( 1 - \sin x ) d x$ is

Suppose S₄ is the upper sum of the area enclosed by the graph $f ( x ) = 2 x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 4 by partitioning [0, 4] into four subintervals [0, 1], [1, 2], [2, 3], and [3, 4]. Then S₄ is

The value of the definite integral $\int _ { 0 } ^ { 2 } \log _ { 2 } 32 d x$ is

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

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

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

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x$ are

The average value of $f ( x ) = \sec ^ { 2 } x$ on $\left[ 0 , \frac { \pi } { 4 } \right]$ is

The indefinite integral $\int \frac { x - 3 } { \sqrt { x ^ { 2 } - 6 x + 5 } } d x$ is

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

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

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