Question 1

Determine the integral by making an appropriate substitution. 
-$\int \frac { 1 } { 4 \sqrt { x } } \sqrt [ 5 ] { 2 + \sqrt { x } }$ dx

Accepted Answer

A)  $\frac { 5 ( 2 + \sqrt { x } ) ^ { 6 / 5 } } { 24 }$ + C 
B)  $\frac { ( 2 + \sqrt { x } ) ^ { 6 } } { 12 }$ + C 
C)  $\frac { 5 ( 2 + \sqrt { x } ) ^ { 6 / 5 } } { 6 }$ + C 
D)  $\frac { 5 ( 2 + \sqrt { x } ) ^ { 6 / 5 } } { 12 }$ + C 
A)  $\frac { 5 ( 2 + \sqrt { x } ) ^ { 6 / 5 } } { 24 }$ + C 
B)  $\frac { ( 2 + \sqrt { x } ) ^ { 6 } } { 12 }$ + C 
C)  $\frac { 5 ( 2 + \sqrt { x } ) ^ { 6 / 5 } } { 6 }$ + C 
D)  $\frac { 5 ( 2 + \sqrt { x } ) ^ { 6 / 5 } } { 12 }$ + C

Question 2

A large conglomerate estimates that a new plant will yield an annual rate of return of $5000 - 250 t$ thousand dollars. If the interest rate is 9%, what is the present value of the income generated in the first three years of the plant's operation?
Enter just an integer (no units) representing the amount to the nearest dollar.

Accepted Answer

The answer of A large conglomerate estimates that a new...

Question 3

Determine the integral by making an appropriate substitution. 
-$\int \sqrt { 2 x ^ { 2 } - 3 x } \left( 3 x - \frac { 9 } { 4 } \right)$ dx

Accepted Answer

A)  $\left( 2 x ^ { 2 } - 3 x \right) ^ { 3 / 2 }$ + C 
B)  $\frac { 1 } { 2 }$ $\left( 2 x ^ { 2 } - 3 x \right) ^ { 3 / 2 }$ + C 
C)  $\left( 2 x ^ { 2 } - 3 x \right) ^ { 2 }$ + C 
D)  $\left( 2 x ^ { 2 } - 3 x \right) ^ { 1 / 2 }$ + C 
E)  none of these 
A)  $\left( 2 x ^ { 2 } - 3 x \right) ^ { 3 / 2 }$ + C 
B)  $\frac { 1 } { 2 }$ $\left( 2 x ^ { 2 } - 3 x \right) ^ { 3 / 2 }$ + C 
C)  $\left( 2 x ^ { 2 } - 3 x \right) ^ { 2 }$ + C 
D)  $\left( 2 x ^ { 2 } - 3 x \right) ^ { 1 / 2 }$ + C 
E)  none of these

Question 4

Which of the following accurately describes integration by parts?

Accepted Answer

A)  $\int f ( x ) g ^ { \prime } ( x ) d x = f ( x ) g ( x ) -$ $\int f ^ { \prime } ( x ) g ( x ) d x$ 
B)  $\int f ( x ) g ^ { \prime } ( x ) d x = f ^ { \prime } ( x ) g ( x ) -$ $\int f ^ { \prime } ( x ) g ( x ) d x$ 
C)  $\int f ( x ) g ( x ) d x = f ^ { \prime } ( x ) g ^ { \prime } ( x ) -$ $\int f ^ { \prime } ( x ) g ^ { \prime } ( x ) d x$ 
D)  $\int f ( x ) g ( x ) d x = f ^ { \prime } ( x ) g ( x ) -$ $\int f ( x ) g ^ { \prime } ( x ) d x$ 
E)  none of these 
A)  $\int f ( x ) g ^ { \prime } ( x ) d x = f ( x ) g ( x ) -$ $\int f ^ { \prime } ( x ) g ( x ) d x$ 
B)  $\int f ( x ) g ^ { \prime } ( x ) d x = f ^ { \prime } ( x ) g ( x ) -$ $\int f ^ { \prime } ( x ) g ( x ) d x$ 
C)  $\int f ( x ) g ( x ) d x = f ^ { \prime } ( x ) g ^ { \prime } ( x ) -$ $\int f ^ { \prime } ( x ) g ^ { \prime } ( x ) d x$ 
D)  $\int f ( x ) g ( x ) d x = f ^ { \prime } ( x ) g ( x ) -$ $\int f ( x ) g ^ { \prime } ( x ) d x$ 
E)  none of these

Question 5

Determine whether the expressions approach a limit as b→∞. If they do, give the value of the limit.   $b ( b + 3 ) ^ { - 2 }$ ,   $\frac { 2 \sqrt { b } - 1 } { \sqrt { b } }$ ,   $\mathrm { e } ^ { 3 b }$ + 2
Enter your answer as just $a , b,c$ where these are either limits (integers) or the words "no limit" in the same order they appear above separated by commas.

Accepted Answer

The answer of Determine whether the expressions approach a limit...

Question 6

Evaluate the integral. 
-$\int _ { 1 } ^ { 2 } x \ln x d x$

Accepted Answer

A)  $\frac { 3 } { 4 }$ + ln 2 
B)  2 ln 2 - 3/4 
C)  2 ln 2 
D)  ln 2 - $\frac { 1 } { 2 }$ 
A)  $\frac { 3 } { 4 }$ + ln 2 
B)  2 ln 2 - 3/4 
C)  2 ln 2 
D)  ln 2 - $\frac { 1 } { 2 }$

Question 7

Find the area under the graph of y = $4 e ^ { - 3 x }$ for x ≥ 1. Enter your answer as a $\mathrm { e } ^ { \mathrm { c } }$ with any fractions reduced of form $\frac { a } { b }$ .

Accepted Answer

The answer of Find the area under the graph of...

Question 8

$\int \cos ^ { 3 }$ x dx   [Hint: $\cos ^ { 2 }$ x = 1 - $\sin ^ { 2 }$ x.]
Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .
No parentheses around arguments of functions.

Accepted Answer

The answer of $\int \cos ^ { 3 }$ x...

Question 9

Decide whether integration by parts or substitution should be used to compute the indefinite integral $\int \ln 4 x d x$ If substitution, indicate the value of u; if by parts, indicate f(x) and g(x).

Accepted Answer

A)  integration by parts; f(x) = 1, g(x) = ln 4x 
B)  substitution; u = 4x 
C)  integration by parts; f(x) = ln 4x, g(x) =1 
D)  integration by parts; f(x) = ln x, g(x) = 4x 
E)  none of these 
A)  integration by parts; f(x) = 1, g(x) = ln 4x 
B)  substitution; u = 4x 
C)  integration by parts; f(x) = ln 4x, g(x) =1 
D)  integration by parts; f(x) = ln x, g(x) = 4x 
E)  none of these

Question 10

$\int \cos x \mathrm { e } ^ { \sin x } \mathrm { dx }$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .
No parentheses around arguments of functions.

Accepted Answer

The answer of \(\int \cos x \mathrm { e }...

Question 11

$\int \sin x \sin ( \cos x ) d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Accepted Answer

The answer of \(\int \sin x \sin ( \cos x...

Question 12

Calculate (a) the trapezoidal approximation and (b) Simpson's approximation to $\int _ { a } ^ { b } f ( x ) d x$ where f is the tabulated function. $$\begin{array} { l r l l l l l l } 
\mathrm { x } & \mathrm { a } = 1.0 & 1.33 & 1.67 & 2.0 & 2.33 & 2.67 & 3.0 = \mathrm { b } \
\hline \mathrm { f } ( \mathrm { x } ) & 5.2 & 6.9 & 1.4 & 0.06 & 2.3 & 0.01 & 1.5 \
\hline
\end{array}$$ Enter your answers in that order as just unlabeled real numbers rounded off to two decimal places, separated by a comma.

Accepted Answer

The answer of Calculate (a) the trapezoidal approximation and (b)...

Question 13

$\int x e ^ { \left( 4 - x ^ { 2 } \right) } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Accepted Answer

The answer of \(\int x e ^ { \left( 4...

Question 14

Determine the integral by making an appropriate substitution. 
-$\int 12 \sin x \sqrt { \cos x }$ dx

Accepted Answer

A)  18 $( \cos x ) ^ { 3 / 2 }$ + C 
B)  8 $( \cos x ) ^ { 3 / 2 }$ + C 
C)  -18 $( \cos x ) ^ { 3 / 2 }$ + C 
D)  -8 $( \cos x ) ^ { 3 / 2 }$ + C 
A)  18 $( \cos x ) ^ { 3 / 2 }$ + C 
B)  8 $( \cos x ) ^ { 3 / 2 }$ + C 
C)  -18 $( \cos x ) ^ { 3 / 2 }$ + C 
D)  -8 $( \cos x ) ^ { 3 / 2 }$ + C

Question 15

Evaluate the integral. 
-$\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ dx

Accepted Answer

A)  $\frac { 728 } { 19,683 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 8 } { 27 }$ 
D)  2 
A)  $\frac { 728 } { 19,683 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 8 } { 27 }$ 
D)  2

Question 16

$\int$ $\frac { 4 x } { \sqrt { x ^ { 2 } + 2 } }$ dx
Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Accepted Answer

The answer of $\int$ \(\frac { 4 x } {...

Question 17

Determine the integral by making an appropriate substitution. 
-$\int 8 x \sqrt [ 3 ] { 4 x ^ { 2 } - 9 }$ dx

Accepted Answer

A)  $\frac { \left( 4 x ^ { 2 } + 9 \right) ^ { 4 } } { 16 }$ + C 
B)  $\frac { \left( 4 x ^ { 2 } - 9 \right) ^ { 4 / 3 } } { 3 }$ + C 
C)  $\frac { 3 \left( 4 x ^ { 2 } + 9 \right) ^ { 4 / 3 } } { 4 }$ + C 
D)  $\frac { 3 \left( 4 x ^ { 2 } - 9 \right) ^ { 4 / 3 } } { 4 }$ + C 
A)  $\frac { \left( 4 x ^ { 2 } + 9 \right) ^ { 4 } } { 16 }$ + C 
B)  $\frac { \left( 4 x ^ { 2 } - 9 \right) ^ { 4 / 3 } } { 3 }$ + C 
C)  $\frac { 3 \left( 4 x ^ { 2 } + 9 \right) ^ { 4 / 3 } } { 4 }$ + C 
D)  $\frac { 3 \left( 4 x ^ { 2 } - 9 \right) ^ { 4 / 3 } } { 4 }$ + C

Question 18

Does this integral $\int x \sec ^ { 2 } x d x$ = x tan x + ln $| \cos x |$ + C?

Accepted Answer

A) True 
 B)False

Determine the integral by making an appropriate substitution. - $\int \frac { 1 } { 4 \sqrt { x } } \sqrt [ 5 ] { 2 + \sqrt { x } }$ dx

Determine the integral by making an appropriate substitution. - $\int \sqrt { 2 x ^ { 2 } - 3 x } \left( 3 x - \frac { 9 } { 4 } \right)$ dx

Which of the following accurately describes integration by parts?

Evaluate the integral. - $\int _ { 1 } ^ { 2 } x \ln x d x$

Find the area under the graph of y = $4 e ^ { - 3 x }$ for x ≥ 1. Enter your answer as a $\mathrm { e } ^ { \mathrm { c } }$ with any fractions reduced of form $\frac { a } { b }$ .

$\int \cos ^ { 3 }$ x dx [Hint: $\cos ^ { 2 }$ x = 1 - $\sin ^ { 2 }$ x.] Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

Decide whether integration by parts or substitution should be used to compute the indefinite integral $\int \ln 4 x d x$ If substitution, indicate the value of u; if by parts, indicate f(x) and g(x).

$\int \cos x \mathrm { e } ^ { \sin x } \mathrm { dx }$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

$\int \sin x \sin ( \cos x ) d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

$\int x e ^ { \left( 4 - x ^ { 2 } \right) } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Determine the integral by making an appropriate substitution. - $\int 12 \sin x \sqrt { \cos x }$ dx

Evaluate the integral. - $\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ dx

$\int$ $\frac { 4 x } { \sqrt { x ^ { 2 } + 2 } }$ dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Determine the integral by making an appropriate substitution. - $\int 8 x \sqrt [ 3 ] { 4 x ^ { 2 } - 9 }$ dx

Does this integral $\int x \sec ^ { 2 } x d x$ = x tan x + ln $| \cos x |$ + C?

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 9: Techniques of Integration

Determine the integral by making an appropriate substitution. - ∫14x2+x5\int \frac { 1 } { 4 \sqrt { x } } \sqrt [ 5 ] { 2 + \sqrt { x } }∫4x​1​52+x​​ dx

Determine the integral by making an appropriate substitution. - ∫2x2−3x(3x−94)\int \sqrt { 2 x ^ { 2 } - 3 x } \left( 3 x - \frac { 9 } { 4 } \right)∫2x2−3x​(3x−49​) dx

Which of the following accurately describes integration by parts?

Evaluate the integral. - ∫12xln⁡xdx\int _ { 1 } ^ { 2 } x \ln x d x∫12​xlnxdx

Find the area under the graph of y = 4e−3x4 e ^ { - 3 x }4e−3x for x ≥ 1. Enter your answer as a ec\mathrm { e } ^ { \mathrm { c } }ec with any fractions reduced of form ab\frac { a } { b }ba​ .

∫cos⁡3\int \cos ^ { 3 }∫cos3 x dx [Hint: cos⁡2\cos ^ { 2 }cos2 x = 1 - sin⁡2\sin ^ { 2 }sin2 x.] Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ . No parentheses around arguments of functions.

Decide whether integration by parts or substitution should be used to compute the indefinite integral ∫ln⁡4xdx\int \ln 4 x d x∫ln4xdx If substitution, indicate the value of u; if by parts, indicate f(x) and g(x).

∫cos⁡xesin⁡xdx\int \cos x \mathrm { e } ^ { \sin x } \mathrm { dx }∫cosxesinxdx Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ . No parentheses around arguments of functions.

∫sin⁡xsin⁡(cos⁡x)dx\int \sin x \sin ( \cos x ) d x∫sinxsin(cosx)dx Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ .

∫xe(4−x2)dx\int x e ^ { \left( 4 - x ^ { 2 } \right) } d x∫xe(4−x2)dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Determine the integral by making an appropriate substitution. - ∫12sin⁡xcos⁡x\int 12 \sin x \sqrt { \cos x }∫12sinxcosx​ dx

Evaluate the integral. - ∫0212x(2x2+1)2\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }∫02​(2x2+1)212x​ dx

∫\int∫ 4xx2+2\frac { 4 x } { \sqrt { x ^ { 2 } + 2 } }x2+2​4x​ dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Determine the integral by making an appropriate substitution. - ∫8x4x2−93\int 8 x \sqrt [ 3 ] { 4 x ^ { 2 } - 9 }∫8x34x2−9​ dx

Does this integral ∫xsec⁡2xdx\int x \sec ^ { 2 } x d x∫xsec2xdx = x tan x + ln ∣cos⁡x∣| \cos x |∣cosx∣ + C?

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Determine the integral by making an appropriate substitution. - $\int \frac { 1 } { 4 \sqrt { x } } \sqrt [ 5 ] { 2 + \sqrt { x } }$ dx

Determine the integral by making an appropriate substitution. - $\int \sqrt { 2 x ^ { 2 } - 3 x } \left( 3 x - \frac { 9 } { 4 } \right)$ dx

Evaluate the integral. - $\int _ { 1 } ^ { 2 } x \ln x d x$

Find the area under the graph of y = $4 e ^ { - 3 x }$ for x ≥ 1. Enter your answer as a $\mathrm { e } ^ { \mathrm { c } }$ with any fractions reduced of form $\frac { a } { b }$ .

$\int \cos ^ { 3 }$ x dx [Hint: $\cos ^ { 2 }$ x = 1 - $\sin ^ { 2 }$ x.] Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

Decide whether integration by parts or substitution should be used to compute the indefinite integral $\int \ln 4 x d x$ If substitution, indicate the value of u; if by parts, indicate f(x) and g(x).

$\int \cos x \mathrm { e } ^ { \sin x } \mathrm { dx }$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

$\int \sin x \sin ( \cos x ) d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

$\int x e ^ { \left( 4 - x ^ { 2 } \right) } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Determine the integral by making an appropriate substitution. - $\int 12 \sin x \sqrt { \cos x }$ dx

Evaluate the integral. - $\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ dx

$\int$ $\frac { 4 x } { \sqrt { x ^ { 2 } + 2 } }$ dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Determine the integral by making an appropriate substitution. - $\int 8 x \sqrt [ 3 ] { 4 x ^ { 2 } - 9 }$ dx

Does this integral $\int x \sec ^ { 2 } x d x$ = x tan x + ln $| \cos x |$ + C?