Question 1

$\int 5 x \sin x ^ { 2 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Accepted Answer

The answer of \(\int 5 x \sin x ^ {...

Question 2

$\int$ $\frac { 1 } { x \ln x \ln ( \ln x ) }$ dx   [Hint: Let u = ln(ln x).]
Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Accepted Answer

The answer of $\int$ \(\frac { 1 } { x...

Question 3

Evaluate the integral using integration by parts.
-$\int \sec ^ { 4 } x d x$

Accepted Answer

A)  $\sec ^ { 2 }$ x tan x + tan x + C 
B)  4 $\sec ^ { 3 }$ x tan x+ C 
C)  $\frac { 1 } { 5 }$ $\sec ^ { 5 }$ x tan x + C 
D)  $\frac { 1 } { 3 }$ $\sec ^ { 2 }$ x tan x + $\frac { 2 } { 3 }$ tan x + C 
A)  $\sec ^ { 2 }$ x tan x + tan x + C 
B)  4 $\sec ^ { 3 }$ x tan x+ C 
C)  $\frac { 1 } { 5 }$ $\sec ^ { 5 }$ x tan x + C 
D)  $\frac { 1 } { 3 }$ $\sec ^ { 2 }$ x tan x + $\frac { 2 } { 3 }$ tan x + C

Question 4

Determine the integral by making an appropriate substitution. 
-$\int \tan ^ { 2 } x \sec ^ { 2 } x d x$   Use the substitution u = tan x.

Accepted Answer

A)  $\frac { \tan ^ { 2 } x } { 2 }$ + C 
B)  sec x tan x + C 
C)  $\frac { \sec ^ { 2 } x } { 2 }$ + C 
D)  $\frac { \tan ^ { 3 } x } { 3 }$ + C 
E)  none of these 
A)  $\frac { \tan ^ { 2 } x } { 2 }$ + C 
B)  sec x tan x + C 
C)  $\frac { \sec ^ { 2 } x } { 2 }$ + C 
D)  $\frac { \tan ^ { 3 } x } { 3 }$ + C 
E)  none of these

Question 5

Does this integral $\int e ^ { x } \cos x d x$ = $\frac { \mathrm { e } ^ { x } \sin x + \mathrm { e } ^ { x } \cos x } { 2 }$ + C?

Accepted Answer

A) True 
 B)False

Question 6

$\int 2 x \left( x ^ { 2 } + 1 \right) ^ { 3 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Accepted Answer

The answer of \(\int 2 x \left( x ^ {...

Question 7

Determine the integral by making an appropriate substitution. 
-$\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }$ dx

Accepted Answer

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

Question 8

Evaluate the integral. 
-$\int _ { 0 } ^ { \pi / 2 } x \cos x d x$

Accepted Answer

A)  1 
B)  $\frac { \pi } { 2 }$ - 1 
C)  0 
D)  π/2 
A)  1 
B)  $\frac { \pi } { 2 }$ - 1 
C)  0 
D)  π/2

Question 9

Calculate (a) the trapezoidal approximation and (b) Simpson's approximation to $\int _ { a } ^ { b } f ( x ) d x$ where f is the tabulated function. $$\begin{array} { l } 
x \quad a = 0 \
\hline f ( x ) \quad 1 \
\hline
\end{array}$$ $$\begin{array} { l l l l l l l l l l } 
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 = b \
\hline 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11
\end{array}$$ Enter your answers in that order as just unlabeled integers separated by a comma.

Accepted Answer

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

Question 10

Evaluate the integral. 
-$\int _ { 1 } ^ { 3 }$ $\frac { 1 } { x }$ dx
Enter just a real number (no approximations and no parentheses around the argument).

Accepted Answer

The answer of Evaluate the integral. 
-\(\int _ { 1...

Question 11

In 1940, the population density of Philadelphia was given by 60 $e ^ { - 0.4 }$ thousand people per square mile at a distance t miles from City Hall. How many people lived between 1 and 3 miles from City Hall?
Enter just an integer (no words) representing the number to the nearest thousand.

Accepted Answer

The answer of In 1940, the population density of Philadelphia...

Question 12

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

Accepted Answer

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

Question 13

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$

Accepted Answer

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

Question 14

Evaluate the integral. 
-$\int _ { - 2 } ^ { 1 } \frac { x } { \sqrt { x + 3 } }$ dx

Accepted Answer

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

Question 15

Does this integral $\int x ^ { 3 } \ln x d x$ = $\frac { 1 } { 4 }$ $x ^ { 4 }$ ln x + $\frac { 1 } { 16 }$ $x ^ { 4 }$ + C?

Accepted Answer

A) True 
 B)False

Question 16

Does this integral $\int _ { 0 } ^ { 1 }$ $\frac { 2 x + 1 } { e ^ { x } }$ dx = $\frac { 3 e - 5 } { e }$ ?

Accepted Answer

A) True 
 B)False

Question 17

Does this integral $\int x e ^ { x } \sin x d x$ = $\frac { - e ^ { x } ( ( x - 1 ) \cos x + x \sin x ) } { 2 }$ + C?

Accepted Answer

A) True 
 B)False

Question 18

$\int$ $\frac { e ^ { x } - e ^ { - x } } { e ^ { x } + e ^ { - x } }$ 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 { e ^ { x...

Question 19

Which of the following is a correct substitution for the integral $\int _ { 1 } ^ { e } \frac { \cos ( \ln x ) } { x } d x$ ?

Accepted Answer

A)  u = ln x; $\int _ { 0 } ^ { 1 } \sin u d u$ 
B)  u = $\frac { \ln x } { x }$ ; $\int _ { 1 } ^ { \mathrm { e } } \cos \mathrm { u } d u$ 
C)  u = ln x; $\int _ { 1 } ^ { \mathrm { e } } \cos \mathrm { u } d u$ 
D)  u= ln x; $\int _ { 0 } ^ { 1 }$ $$\cos u$$
U du 
E)  u = ln x; $\int _ { 0 } ^ { 1 } \cos u \mathrm { du }$ 
A)  u = ln x; $\int _ { 0 } ^ { 1 } \sin u d u$ 
B)  u = $\frac { \ln x } { x }$ ; $\int _ { 1 } ^ { \mathrm { e } } \cos \mathrm { u } d u$ 
C)  u = ln x; $\int _ { 1 } ^ { \mathrm { e } } \cos \mathrm { u } d u$ 
D)  u= ln x; $\int _ { 0 } ^ { 1 }$ $$\cos u$$
U du 
E)  u = ln x; $\int _ { 0 } ^ { 1 } \cos u \mathrm { du }$

Question 20

Determine the integral by making an appropriate substitution. 
-$\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }$

Accepted Answer

A)  $\mathrm { e } ^ { 2 x }$ + C 
B)  $\left( 1 + 2 e ^ { x } \right) ^ { 2 }$ + C 
C)  ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C 
D)  $\frac { 1 } { 2 }$ ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C 
E)  none of these 
A)  $\mathrm { e } ^ { 2 x }$ + C 
B)  $\left( 1 + 2 e ^ { x } \right) ^ { 2 }$ + C 
C)  ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C 
D)  $\frac { 1 } { 2 }$ ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C 
E)  none of these

$\int 5 x \sin x ^ { 2 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

$\int$ $\frac { 1 } { x \ln x \ln ( \ln x ) }$ dx [Hint: Let u = ln(ln x).] Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Evaluate the integral using integration by parts. - $\int \sec ^ { 4 } x d x$

Determine the integral by making an appropriate substitution. - $\int \tan ^ { 2 } x \sec ^ { 2 } x d x$ Use the substitution u = tan x.

Does this integral $\int e ^ { x } \cos x d x$ = $\frac { \mathrm { e } ^ { x } \sin x + \mathrm { e } ^ { x } \cos x } { 2 }$ + C?

$\int 2 x \left( x ^ { 2 } + 1 \right) ^ { 3 } d x$ 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. - $\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }$ dx

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } x \cos x d x$

Calculate (a) the trapezoidal approximation and (b) Simpson's approximation to $\int _ { a } ^ { b } f ( x ) d x$ where f is the tabulated function. xa=0 f(x)1 1 2 3 4 5 6 7 8 9 10=b 2 3 4 5 6 7 8 9 10 11 Enter your answers in that order as just unlabeled integers separated by a comma.

Evaluate the integral. - $\int _ { 1 } ^ { 3 }$ $\frac { 1 } { x }$ dx Enter just a real number (no approximations and no parentheses around the argument).

In 1940, the population density of Philadelphia was given by 60 $e ^ { - 0.4 }$ thousand people per square mile at a distance t miles from City Hall. How many people lived between 1 and 3 miles from City Hall? Enter just an integer (no words) representing the number to the nearest thousand.

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

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$

Evaluate the integral. - $\int _ { - 2 } ^ { 1 } \frac { x } { \sqrt { x + 3 } }$ dx

Does this integral $\int x ^ { 3 } \ln x d x$ = $\frac { 1 } { 4 }$ $x ^ { 4 }$ ln x + $\frac { 1 } { 16 }$ $x ^ { 4 }$ + C?

Does this integral $\int _ { 0 } ^ { 1 }$ $\frac { 2 x + 1 } { e ^ { x } }$ dx = $\frac { 3 e - 5 } { e }$ ?

Does this integral $\int x e ^ { x } \sin x d x$ = $\frac { - e ^ { x } ( ( x - 1 ) \cos x + x \sin x ) } { 2 }$ + C?

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

Which of the following is a correct substitution for the integral $\int _ { 1 } ^ { e } \frac { \cos ( \ln x ) } { x } d x$ ?

Determine the integral by making an appropriate substitution. - $\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }$

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

∫5xsin⁡x2dx\int 5 x \sin x ^ { 2 } d x∫5xsinx2dx Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ .

∫\int∫ 1xln⁡xln⁡(ln⁡x)\frac { 1 } { x \ln x \ln ( \ln x ) }xlnxln(lnx)1​ dx [Hint: Let u = ln(ln x).] Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ .

Evaluate the integral using integration by parts. - ∫sec⁡4xdx\int \sec ^ { 4 } x d x∫sec4xdx

Determine the integral by making an appropriate substitution. - ∫tan⁡2xsec⁡2xdx\int \tan ^ { 2 } x \sec ^ { 2 } x d x∫tan2xsec2xdx Use the substitution u = tan x.

Does this integral ∫excos⁡xdx\int e ^ { x } \cos x d x∫excosxdx = exsin⁡x+excos⁡x2\frac { \mathrm { e } ^ { x } \sin x + \mathrm { e } ^ { x } \cos x } { 2 }2exsinx+excosx​ + C?

∫2x(x2+1)3dx\int 2 x \left( x ^ { 2 } + 1 \right) ^ { 3 } d x∫2x(x2+1)3dx 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. - ∫x+32x2+3x\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }∫x2+3xx+23​​ dx

Evaluate the integral. - ∫0π/2xcos⁡xdx\int _ { 0 } ^ { \pi / 2 } x \cos x d x∫0π/2​xcosxdx

Evaluate the integral. - ∫13\int _ { 1 } ^ { 3 }∫13​ 1x\frac { 1 } { x }x1​ dx Enter just a real number (no approximations and no parentheses around the argument).

Evaluate the integral. - ∫12−e1/xx2dx\int _ { 1 } ^ { 2 } - \frac { e ^ { 1 / x } } { x ^ { 2 } } d x∫12​−x2e1/x​dx

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - ∫2∞dxxln⁡x\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }∫2∞​xlnxdx​

Evaluate the integral. - ∫−21xx+3\int _ { - 2 } ^ { 1 } \frac { x } { \sqrt { x + 3 } }∫−21​x+3​x​ dx

Does this integral ∫x3ln⁡xdx\int x ^ { 3 } \ln x d x∫x3lnxdx = 14\frac { 1 } { 4 }41​ x4x ^ { 4 }x4 ln x + 116\frac { 1 } { 16 }161​ x4x ^ { 4 }x4 + C?

Does this integral ∫01\int _ { 0 } ^ { 1 }∫01​ 2x+1ex\frac { 2 x + 1 } { e ^ { x } }ex2x+1​ dx = 3e−5e\frac { 3 e - 5 } { e }e3e−5​ ?

Does this integral ∫xexsin⁡xdx\int x e ^ { x } \sin x d x∫xexsinxdx = −ex((x−1)cos⁡x+xsin⁡x)2\frac { - e ^ { x } ( ( x - 1 ) \cos x + x \sin x ) } { 2 }2−ex((x−1)cosx+xsinx)​ + C?

∫\int∫ ex−e−xex+e−x\frac { e ^ { x } - e ^ { - x } } { e ^ { x } + e ^ { - x } }ex+e−xex−e−x​ dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Which of the following is a correct substitution for the integral ∫1ecos⁡(ln⁡x)xdx\int _ { 1 } ^ { e } \frac { \cos ( \ln x ) } { x } d x∫1e​xcos(lnx)​dx ?

Determine the integral by making an appropriate substitution. - ∫ex1+2exdx\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }∫1+2exex​dx

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

$\int 5 x \sin x ^ { 2 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

$\int$ $\frac { 1 } { x \ln x \ln ( \ln x ) }$ dx [Hint: Let u = ln(ln x).] Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Evaluate the integral using integration by parts. - $\int \sec ^ { 4 } x d x$

Determine the integral by making an appropriate substitution. - $\int \tan ^ { 2 } x \sec ^ { 2 } x d x$ Use the substitution u = tan x.

Does this integral $\int e ^ { x } \cos x d x$ = $\frac { \mathrm { e } ^ { x } \sin x + \mathrm { e } ^ { x } \cos x } { 2 }$ + C?

$\int 2 x \left( x ^ { 2 } + 1 \right) ^ { 3 } d x$ 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. - $\int \frac { x + \frac { 3 } { 2 } } { x ^ { 2 } + 3 x }$ dx

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } x \cos x d x$

Evaluate the integral. - $\int _ { 1 } ^ { 3 }$ $\frac { 1 } { x }$ dx Enter just a real number (no approximations and no parentheses around the argument).

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

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 2 } ^ { \infty } \frac { d x } { x \ln x }$

Evaluate the integral. - $\int _ { - 2 } ^ { 1 } \frac { x } { \sqrt { x + 3 } }$ dx

Does this integral $\int x ^ { 3 } \ln x d x$ = $\frac { 1 } { 4 }$ $x ^ { 4 }$ ln x + $\frac { 1 } { 16 }$ $x ^ { 4 }$ + C?

Does this integral $\int _ { 0 } ^ { 1 }$ $\frac { 2 x + 1 } { e ^ { x } }$ dx = $\frac { 3 e - 5 } { e }$ ?

Does this integral $\int x e ^ { x } \sin x d x$ = $\frac { - e ^ { x } ( ( x - 1 ) \cos x + x \sin x ) } { 2 }$ + C?

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

Which of the following is a correct substitution for the integral $\int _ { 1 } ^ { e } \frac { \cos ( \ln x ) } { x } d x$ ?

Determine the integral by making an appropriate substitution. - $\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }$