Question 1

Find the indefinite integral. $$\int \sqrt { 49 - 4 x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 49 } { 4 } \sin ^ { - 1 } \left( \frac { 2 x } { 7 } \right) + \frac { 2 x \sqrt { 49 - 4 x ^ { 2 } } } { 3 } + C$ 
B)  $\frac { 49 } { 4 } \sin ^ { - 1 } \left( \frac { 2 x } { 7 } \right) - \frac { x \sqrt { 49 - 4 x ^ { 2 } } } { 3 } + C$ 
C) $\frac { 49 } { 4 } \sin ^ { - 1 } \left( \frac { 2 x } { 7 } \right) + \frac { x \sqrt { 49 - 4 x ^ { 2 } } } { 2 } + C$ 
D) $- \frac { 49 } { 4 } \cos ^ { - 1 } \left( \frac { 2 x } { 7 } \right) + \frac { x \sqrt { 49 - 4 x ^ { 2 } } } { 2 } + C$ 
E) $- \frac { 49 } { 4 } \cos ^ { - 1 } \left( \frac { 2 x } { 7 } \right) - \frac { 3 x \sqrt { 49 - 4 x ^ { 2 } } } { 2 } + C$ 
A)  $\frac { 49 } { 4 } \sin ^ { - 1 } \left( \frac { 2 x } { 7 } \right) + \frac { 2 x \sqrt { 49 - 4 x ^ { 2 } } } { 3 } + C$ 
B)  $\frac { 49 } { 4 } \sin ^ { - 1 } \left( \frac { 2 x } { 7 } \right) - \frac { x \sqrt { 49 - 4 x ^ { 2 } } } { 3 } + C$ 
C) $\frac { 49 } { 4 } \sin ^ { - 1 } \left( \frac { 2 x } { 7 } \right) + \frac { x \sqrt { 49 - 4 x ^ { 2 } } } { 2 } + C$ 
D) $- \frac { 49 } { 4 } \cos ^ { - 1 } \left( \frac { 2 x } { 7 } \right) + \frac { x \sqrt { 49 - 4 x ^ { 2 } } } { 2 } + C$ 
E) $- \frac { 49 } { 4 } \cos ^ { - 1 } \left( \frac { 2 x } { 7 } \right) - \frac { 3 x \sqrt { 49 - 4 x ^ { 2 } } } { 2 } + C$

Question 2

$\text { Evaluate the limit } \lim _ { x \rightarrow 10 } \frac { - 9 ( x - 10 ) } { x ^ { 2 } - 100 } \text { first by using techniques from Chapter } 1 \text { then }$ by using L'Hopital's Rule.

Accepted Answer

A)  $\frac { 9 } { 20 }$ 
B)  $- \frac { 9 } { 100 }$ 
C)  $- \frac { 9 } { 20 }$ 
D)  0 
E)  does not exist 
A)  $\frac { 9 } { 20 }$ 
B)  $- \frac { 9 } { 100 }$ 
C)  $- \frac { 9 } { 20 }$ 
D)  0 
E)  does not exist

Question 3

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { x } + 5 x \right) ^ { 5 / x } \text { using L'Hopital's Rule if necessary. }$

Accepted Answer

A)  $e ^ { 5 }$ 
B)  $e ^ { 10 }$ 
C)  1 
D)  $e ^ { 30 }$ 
E)  $e + 5$ 
A)  $e ^ { 5 }$ 
B)  $e ^ { 10 }$ 
C)  1 
D)  $e ^ { 30 }$ 
E)  $e + 5$

Question 4

$\text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. }$

Accepted Answer

A)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 20 } \csc ^ { 20 } x + C$ 
B)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$ 
C)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 20 } \csc ^ { 20 x } x + C$ 
D)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 20 } \csc ^ { 20 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$ 
E)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 21 } \csc ^ { 21 } x + C$ 
A)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 20 } \csc ^ { 20 } x + C$ 
B)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$ 
C)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 20 } \csc ^ { 20 x } x + C$ 
D)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 20 } \csc ^ { 20 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$ 
E)  $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 21 } \csc ^ { 21 } x + C$

Question 5

$\text { Find the indefinite integral by making the substitution } x = 6 \sec \theta \text {. }$ $$\int \frac { \sqrt { x ^ { 2 } - 36 } } { x } d x$$

Accepted Answer

A)  $\sqrt { x ^ { 2 } - 36 } - 6 \sec ^ { - 1 } \left( \frac { x } { 6 } \right) + C$ 
B)  $\sqrt { x ^ { 2 } - 36 } + 6 \sec ^ { - 1 } \left( \frac { x } { 6 } \right) + C$ 
C)  $\sqrt { x ^ { 2 } - 36 } + 6 \sec ^ { - 1 } \left( \frac { 6 } { x } \right) + C$ 
D)  $\sqrt { x ^ { 2 } - 36 } - 36 \sec ^ { - 1 } \left( \frac { x } { 6 } \right) + C$ 
E)  $\sqrt { x ^ { 2 } - 36 } - 6 \sec ^ { - 1 } \left( \frac { 6 } { x } \right) + C$ 
A)  $\sqrt { x ^ { 2 } - 36 } - 6 \sec ^ { - 1 } \left( \frac { x } { 6 } \right) + C$ 
B)  $\sqrt { x ^ { 2 } - 36 } + 6 \sec ^ { - 1 } \left( \frac { x } { 6 } \right) + C$ 
C)  $\sqrt { x ^ { 2 } - 36 } + 6 \sec ^ { - 1 } \left( \frac { 6 } { x } \right) + C$ 
D)  $\sqrt { x ^ { 2 } - 36 } - 36 \sec ^ { - 1 } \left( \frac { x } { 6 } \right) + C$ 
E)  $\sqrt { x ^ { 2 } - 36 } - 6 \sec ^ { - 1 } \left( \frac { 6 } { x } \right) + C$

Question 6

Find the indefinite integral. $\int \cos ^ { 4 } 5 x d x$

Accepted Answer

A)  $\frac { 60 x + 8 \sin ( 10 ) x + \sin ( 15 ) x } { 160 } + C$ 
B)  $\frac { 60 x + 8 \sin ( 5 ) x + \sin ( 20 ) x } { 160 } + C$ 
C)  $\frac { 20 x + 4 \sin ( 10 ) x + \sin ( 15 ) x } { 160 } + C$ 
D)  $\frac { 60 x + 8 \sin ( 10 ) x + \sin ( 20 ) x } { 160 } + C$ 
E)  $\frac { 20 x + 8 \sin ( 10 ) x + \sin ( 20 ) x } { 160 } + C$ 
A)  $\frac { 60 x + 8 \sin ( 10 ) x + \sin ( 15 ) x } { 160 } + C$ 
B)  $\frac { 60 x + 8 \sin ( 5 ) x + \sin ( 20 ) x } { 160 } + C$ 
C)  $\frac { 20 x + 4 \sin ( 10 ) x + \sin ( 15 ) x } { 160 } + C$ 
D)  $\frac { 60 x + 8 \sin ( 10 ) x + \sin ( 20 ) x } { 160 } + C$ 
E)  $\frac { 20 x + 8 \sin ( 10 ) x + \sin ( 20 ) x } { 160 } + C$

Question 7

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 7 x - 6 } { x \left( x ^ { 2 } + 10 \right) ^ { 2 } }$

Accepted Answer

A)  $\frac { A } { x } + \frac { B x + C } { \left( x ^ { 2 } + 10 \right) ^ { 2 } }$ 
B) $\frac { A } { x } + \frac { B } { x ^ { 2 } + 10 } + \frac { C } { \left( x ^ { 2 } + 10 \right) ^ { 2 } }$ 
C)  $\frac { A } { x } + \frac { B } { x + 10 } + \frac { C } { x - 10 }$ 
D)  $\frac { A } { x } + \frac { B } { x + 10 } + \frac { C } { ( x + 10 ) ^ { 2 } } + \frac { D } { x - 10 } + \frac { E } { ( x - 10 ) ^ { 2 } }$ 
E) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 10 } + \frac { D x + E } { \left( x ^ { 2 } + 10 \right) ^ { 2 } }$ 
A)  $\frac { A } { x } + \frac { B x + C } { \left( x ^ { 2 } + 10 \right) ^ { 2 } }$ 
B) $\frac { A } { x } + \frac { B } { x ^ { 2 } + 10 } + \frac { C } { \left( x ^ { 2 } + 10 \right) ^ { 2 } }$ 
C)  $\frac { A } { x } + \frac { B } { x + 10 } + \frac { C } { x - 10 }$ 
D)  $\frac { A } { x } + \frac { B } { x + 10 } + \frac { C } { ( x + 10 ) ^ { 2 } } + \frac { D } { x - 10 } + \frac { E } { ( x - 10 ) ^ { 2 } }$ 
E) $\frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 10 } + \frac { D x + E } { \left( x ^ { 2 } + 10 \right) ^ { 2 } }$

Find the indefinite integral. $\int \sqrt { 49 - 4 x ^ { 2 } } d x$

$\text { Evaluate the limit } \lim _ { x \rightarrow 10 } \frac { - 9 ( x - 10 ) } { x ^ { 2 } - 100 } \text { first by using techniques from Chapter } 1 \text { then }$ by using L'Hopital's Rule.

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { x } + 5 x \right) ^ { 5 / x } \text { using L'Hopital's Rule if necessary. }$

$\text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. }$

$\text { Find the indefinite integral by making the substitution } x = 6 \sec \theta \text {. }$ $\int \frac { \sqrt { x ^ { 2 } - 36 } } { x } d x$

Find the indefinite integral. $\int \cos ^ { 4 } 5 x d x$

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 7 x - 6 } { x \left( x ^ { 2 } + 10 \right) ^ { 2 } }$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 8: Basic Integration Rules

Find the indefinite integral. ∫49−4x2dx\int \sqrt { 49 - 4 x ^ { 2 } } d x∫49−4x2​dx

Find ∫cot⁡3xcsc⁡19xdx. \text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. } Find ∫cot3xcsc19xdx.

Find the indefinite integral. ∫cos⁡45xdx\int \cos ^ { 4 } 5 x d x∫cos45xdx

Write the form of the partial fraction decomposition for the following rational expression. 7x−6x(x2+10)2\frac { 7 x - 6 } { x \left( x ^ { 2 } + 10 \right) ^ { 2 } }x(x2+10)27x−6​

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the indefinite integral. $\int \sqrt { 49 - 4 x ^ { 2 } } d x$

$\text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. }$

Find the indefinite integral. $\int \cos ^ { 4 } 5 x d x$

Write the form of the partial fraction decomposition for the following rational expression. $\frac { 7 x - 6 } { x \left( x ^ { 2 } + 10 \right) ^ { 2 } }$