Question 1

Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$.

Accepted Answer

A)  $\frac { x ^ { 3 } - 25 x } { x ^ { 3 } + 25 x } + C$ 
B)  $x - 10 \arctan \left( \frac { x } { 5 } \right) + C$ 
C)  $x - 6 \arctan \left( \frac { x } { 5 } \right) + C$ 
D)  $\frac { x ^ { 3 } - 75 x } { x ^ { 3 } + 75 x } + C$ 
A)  $\frac { x ^ { 3 } - 25 x } { x ^ { 3 } + 25 x } + C$ 
B)  $x - 10 \arctan \left( \frac { x } { 5 } \right) + C$ 
C)  $x - 6 \arctan \left( \frac { x } { 5 } \right) + C$ 
D)  $\frac { x ^ { 3 } - 75 x } { x ^ { 3 } + 75 x } + C$

Question 2

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c k$.
$f ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }$ is the left endpoint

Accepted Answer

The answer of Use the definition of area to find...

Question 3

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 4

Find $g ^ { \prime } ( x )$ by evaluating the integral using Part 2 of the Fundamental Theorem and then differentiating.
$$g ( x ) = \int _ { x } ^ { x } ( 6 + \cos t ) d t$$

Accepted Answer

The answer of Find \(g ^ { \prime } (...

Question 5

Use the following property of the definite integral to estimate the definite integral $\int _ { x / 6 } ^ { s / 4 } 6 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$, then 
$$m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$$

Accepted Answer

A) 
$\frac { \pi } { 2 } \leq \int _ { x / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$ 
B)  $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { x / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$ 
C) 
$\frac { \pi } { 4 } \leq \int _ { x / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$ 
D)  $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { z / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$ 
A) 
$\frac { \pi } { 2 } \leq \int _ { x / 6 } ^ { x / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 2 }$ 
B)  $\frac { \sqrt { 2 } \pi } { 4 } \leq \int _ { x / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 4 }$ 
C) 
$\frac { \pi } { 4 } \leq \int _ { x / 6 } ^ { \pi / 4 } 6 \sin x d x \leq \frac { \sqrt { 2 } \pi } { 4 }$ 
D)  $\frac { \sqrt { 2 } \pi } { 2 } \leq \int _ { z / 6 } ^ { z / 4 } 6 \sin x d x \leq \frac { \pi } { 2 }$

Question 6

Evaluate the integral. $$\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$$

Accepted Answer

A)  $26.4$ 
B)  $31.4$ 
C)  $46.4$ 
D)  $36.4$ 
E)  $41.4$ 
A)  $26.4$ 
B)  $31.4$ 
C)  $46.4$ 
D)  $36.4$ 
E)  $41.4$

Question 7

The speed of a runner increased steadily during the first three seconds of a race.Her speed at half-second intervals is given in the table.Find a lower estimate for the distance that she traveled
during these three seconds. $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline t ( \mathrm {~s} ) & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \
\hline ( \mathrm { f } / \mathrm { s } ) & 0 & 2.1 & 5 & 6.8 & 9.7 & 12.9 & 17.5 \
\hline
\end{array}$$

Accepted Answer

The answer of The speed of a runner increased steadily...

Question 8

Evaluate the integral $\int _ { 1 } ^ { 9 } \frac { 8 x ^ { 2 } - 3 x + 7 } { x } d x$

Accepted Answer

A)  $616 + 7 \ln 9$ 
B)  $296 + 7 \ln 9$ 
C)  $616 + 9 \ln 7$ 
D)  $296 + 9 \ln 7$ 
A)  $616 + 7 \ln 9$ 
B)  $296 + 7 \ln 9$ 
C)  $616 + 9 \ln 7$ 
D)  $296 + 9 \ln 7$

Question 9

Evaluate the integral. $$\int _ { 0 } ^ { 1 } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x$$

Accepted Answer

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

Question 10

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$.
$f ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }$ is the right endpoint

Accepted Answer

The answer of Use the definition of area to find...

Question 11

Use the Midpoint Rule with $n = 6$ to approximate the integral. Round the answer to 3 decimal places. $$\int _ { 0 } ^ { 12 } 2 \sin \sqrt { t } d t$$

Accepted Answer

The answer of Use the Midpoint Rule with \(n =...

Question 12

Evaluate the integral. $$\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }$$

Accepted Answer

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

Question 13

Find the indefinite integral.
$$\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$$

Accepted Answer

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

Question 14

Find the integral using an appropriate trigonometric substitution. $$\int x \sqrt { 9 - x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
B)  $- \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
C)  $- \frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
D)  $\frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
A)  $\frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
B)  $- \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
C)  $- \frac { 1 } { 3 } x ^ { 2 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$ 
D)  $\frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right) ^ { 3 / 2 } + C$

Question 15

Find the integral.
$$\int \sin ^ { 3 } x \cos ^ { 6 } x d x$$

Accepted Answer

A)  $\frac { 1 } { 7 } \cos ^ { 7 } x - \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
B)  $- \frac { 1 } { 9 } \cos ^ { 9 } x + \frac { 1 } { 7 } \cos ^ { 7 } x + C$ 
C)  $- \frac { 1 } { 7 } \cos ^ { 7 } x + \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
D)  $\frac { 1 } { 9 } \cos ^ { 9 } x - \frac { 1 } { 7 } \cos ^ { 7 } x + C$ 
A)  $\frac { 1 } { 7 } \cos ^ { 7 } x - \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
B)  $- \frac { 1 } { 9 } \cos ^ { 9 } x + \frac { 1 } { 7 } \cos ^ { 7 } x + C$ 
C)  $- \frac { 1 } { 7 } \cos ^ { 7 } x + \frac { 1 } { 5 } \cos ^ { 5 } x + C$ 
D)  $\frac { 1 } { 9 } \cos ^ { 9 } x - \frac { 1 } { 7 } \cos ^ { 7 } x + C$

Question 16

Find the integral using the indicated substitution. $$\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$$

Accepted Answer

A)  $\frac { 1 } { 28 } ( 4 x + 3 ) ^ { 7 } + C$ 
B)  $6 ( 4 x + 3 ) ^ { 7 } + C$ 
C)  $\frac { 1 } { 6 } ( 4 x + 3 ) ^ { 7 } + C$ 
D)  $\frac { 1 } { 7 } ( 4 x + 3 ) ^ { 7 } + C$ 
A)  $\frac { 1 } { 28 } ( 4 x + 3 ) ^ { 7 } + C$ 
B)  $6 ( 4 x + 3 ) ^ { 7 } + C$ 
C)  $\frac { 1 } { 6 } ( 4 x + 3 ) ^ { 7 } + C$ 
D)  $\frac { 1 } { 7 } ( 4 x + 3 ) ^ { 7 } + C$

Question 17

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

Accepted Answer

A) 
$- \frac { 2 x } { \left( x ^ { 2 } + 2 \right) ^ { 2 } }$ 
B)  $- \frac { 2 } { 9 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { x ^ { 2 } + 2 }$ 
A) 
$- \frac { 2 x } { \left( x ^ { 2 } + 2 \right) ^ { 2 } }$ 
B)  $- \frac { 2 } { 9 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { x ^ { 2 } + 2 }$

Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$ .

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c k$ . $f ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }$ is the left endpoint

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

Find $g ^ { \prime } ( x )$ by evaluating the integral using Part 2 of the Fundamental Theorem and then differentiating. $g ( x ) = \int _ { x } ^ { x } ( 6 + \cos t ) d t$

Use the following property of the definite integral to estimate the definite integral $\int _ { x / 6 } ^ { s / 4 } 6 \sin x d x$ : If $m \leq f ( x ) \leq M$ on $[ a , b ]$ , then $m ( b - a ) \leq \int _ { a } ^ { b } f ( x ) d x \leq M ( b - a )$

Evaluate the integral. $\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$

The speed of a runner increased steadily during the first three seconds of a race.Her speed at half-second intervals is given in the table.Find a lower estimate for the distance that she traveled during these three seconds. t() 0 0.5 1.0 1.5 2.0 2.5 3.0 (/) 0 2.1 5 6.8 9.7 12.9 17.5

Evaluate the integral $\int _ { 1 } ^ { 9 } \frac { 8 x ^ { 2 } - 3 x + 7 } { x } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x$

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$ . $f ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }$ is the right endpoint

Use the Midpoint Rule with $n = 6$ to approximate the integral. Round the answer to 3 decimal places. $\int _ { 0 } ^ { 12 } 2 \sin \sqrt { t } d t$

Evaluate the integral. $\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }$

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 9 - x ^ { 2 } } d x$

Find the integral. $\int \sin ^ { 3 } x \cos ^ { 6 } x d x$

Find the integral using the indicated substitution. $\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$

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

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

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Exam 4: Applications of Differentiation

Find the indefinite integral ∫x2−25x2+25dx\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x∫x2+25x2−25​dx .

Use the definition of area to find the area of the region under the graph of fff on [a,b][ a , b ][a,b] using the indicated choice of ckc kck . f(x)=x2,[0,6],ckf ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }f(x)=x2,[0,6],ck​ is the left endpoint

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

Find g′(x)g ^ { \prime } ( x )g′(x) by evaluating the integral using Part 2 of the Fundamental Theorem and then differentiating. g(x)=∫xx(6+cos⁡t)dtg ( x ) = \int _ { x } ^ { x } ( 6 + \cos t ) d tg(x)=∫xx​(6+cost)dt

Evaluate the integral. ∫14x2+7xdx\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x∫14​x​x2+7​dx

The speed of a runner increased steadily during the first three seconds of a race.Her speed at half-second intervals is given in the table.Find a lower estimate for the distance that she traveled during these three seconds. t() 0 0.5 1.0 1.5 2.0 2.5 3.0 (/) 0 2.1 5 6.8 9.7 12.9 17.5

Evaluate the integral ∫198x2−3x+7xdx\int _ { 1 } ^ { 9 } \frac { 8 x ^ { 2 } - 3 x + 7 } { x } d x∫19​x8x2−3x+7​dx

Evaluate the integral. ∫01e2x1+e4xdx\int _ { 0 } ^ { 1 } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x∫01​1+e4xe2x​dx

Use the definition of area to find the area of the region under the graph of fff on [a,b][ a , b ][a,b] using the indicated choice of ckc _ { k }ck​ . f(x)=x2+5x+1,[−1,1],ckf ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }f(x)=x2+5x+1,[−1,1],ck​ is the right endpoint

Use the Midpoint Rule with n=6n = 6n=6 to approximate the integral. Round the answer to 3 decimal places. ∫0122sin⁡tdt\int _ { 0 } ^ { 12 } 2 \sin \sqrt { t } d t∫012​2sint​dt

Evaluate the integral. ∫0x/610+cos⁡2θcos⁡2θ\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }∫0x/6​cos2θ10+cos2θ​

Find the indefinite integral. ∫x−2/3x1/3−2dx\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x∫x−2/3x1/3−2​dx

Find the integral using an appropriate trigonometric substitution. ∫x9−x2dx\int x \sqrt { 9 - x ^ { 2 } } d x∫x9−x2​dx

Find the integral. ∫sin⁡3xcos⁡6xdx\int \sin ^ { 3 } x \cos ^ { 6 } x d x∫sin3xcos6xdx

Find the integral using the indicated substitution. ∫(4x+3)6dx,u=4x+3\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3∫(4x+3)6dx,u=4x+3

Find the derivative of the function. g(x)=∫1x1t2+2dtg ( x ) = \int _ { 1 } ^ { x } \frac { 1 } { t ^ { 2 } + 2 } d tg(x)=∫1x​t2+21​dt

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

Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$ .

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c k$ . $f ( x ) = x ^ { 2 } , \quad [ 0,6 ] , \quad c _ { k }$ is the left endpoint

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

Find $g ^ { \prime } ( x )$ by evaluating the integral using Part 2 of the Fundamental Theorem and then differentiating. $g ( x ) = \int _ { x } ^ { x } ( 6 + \cos t ) d t$

Evaluate the integral. $\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$

Evaluate the integral $\int _ { 1 } ^ { 9 } \frac { 8 x ^ { 2 } - 3 x + 7 } { x } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x$

Use the definition of area to find the area of the region under the graph of $f$ on $[ a , b ]$ using the indicated choice of $c _ { k }$ . $f ( x ) = x ^ { 2 } + 5 x + 1 , \quad [ - 1,1 ] , \quad c _ { k }$ is the right endpoint

Use the Midpoint Rule with $n = 6$ to approximate the integral. Round the answer to 3 decimal places. $\int _ { 0 } ^ { 12 } 2 \sin \sqrt { t } d t$

Evaluate the integral. $\int _ { 0 } ^ { x / 6 } \frac { 10 + \cos ^ { 2 } \theta } { \cos ^ { 2 } \theta }$

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 9 - x ^ { 2 } } d x$

Find the integral. $\int \sin ^ { 3 } x \cos ^ { 6 } x d x$

Find the integral using the indicated substitution. $\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$

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