Question 1

Determine the integral by making an appropriate substitution. 
-$\int x ^ { 2 } \sin \left( 7 x ^ { 3 } - 6 \right) d x$

Accepted Answer

A)  21 cos(7 $x ^ { 3 }$ - 6) + C 
B)  - $\frac { 1 } { 21 }$ cos(7 $x ^ { 3 }$ - 6) + C 
C)  $\frac { 1 } { 21 }$ cos(7 $x ^ { 3 }$ - 6) + C 
D)  -21 cos(7 $x ^ { 3 }$ - 6) + C 
A)  21 cos(7 $x ^ { 3 }$ - 6) + C 
B)  - $\frac { 1 } { 21 }$ cos(7 $x ^ { 3 }$ - 6) + C 
C)  $\frac { 1 } { 21 }$ cos(7 $x ^ { 3 }$ - 6) + C 
D)  -21 cos(7 $x ^ { 3 }$ - 6) + C

Question 2

The shaded area in the diagram represents an estimation $\int _ { a } ^ { b } f ( x )$ dx using:

Accepted Answer

A)  Simpson's rule with n =4 
B)  a Riemann Sum using right endpoints 
C)  the midpoint rule with n = 4 
D)  a Riemann Sum using left endpoints 
E)  the trapezoid rule with n = 4 
A)  Simpson's rule with n =4 
B)  a Riemann Sum using right endpoints 
C)  the midpoint rule with n = 4 
D)  a Riemann Sum using left endpoints 
E)  the trapezoid rule with n = 4

Question 3

Does this integral $\int x \cos 5 x d x$ = $\frac { 1 } { 5 }$ x sin 5x - $\frac { 1 } { 25 }$ cos 5x + C?

Accepted Answer

A) True 
 B)False

Question 4

Approximate $\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } }$ dx; n = 4, by (a) Simpson's rule and (b) the trapezoidal rule.
Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

Accepted Answer

The answer of Approximate \(\int _ { 0 } ^...

Question 5

Does this integral $\int$ $\frac { \ln ( \ln x ) } { x }$ dx = ln x(ln(ln x) -1) + C?

Accepted Answer

A) True 
 B)False

Question 6

Determine the integral by making an appropriate substitution. 
-$\int \frac { \sin \theta \cos ^ { 2 } \theta } { 1 + \cos ^ { 3 } \theta }$ dθ

Accepted Answer

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

Question 7

Determine the integral by making an appropriate substitution. 
-$\int \frac { \sin x } { 1 - \cos x } d x$

Accepted Answer

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

Question 8

$\int ( x - 1 ) e ^ { \left( 3 x ^ { 2 } - 6 x \right) } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Accepted Answer

The answer of \(\int ( x - 1 ) e...

Question 9

Does this integral $\int$ $\frac { \cos x } { ( 3 \sin x + 1 ) ^ { 2 } }$ dx = $\frac { 1 } { 3 ( 3 \sin x + 1 ) }$ + C?

Accepted Answer

A) True 
 B)False

Question 10

Evaluate the integral using integration by parts.
-$\int e ^ { a x } \sin b x d x$

Accepted Answer

A)  $\mathrm { e } ^ { \mathrm { ax } }$ (sin bx - cos bx) + C 
B)  $\mathrm { e } ^ { \mathrm { ax } }$ (sin bx + cos bx) + C 
C)  $\frac { 1 } { 2 b }$ $\mathrm { e } ^ { \mathrm { ax } }$ $\cos ^ { 2 }$ bx + C 
D)  $\mathrm { e } ^ { \mathrm { ax } }$ $\left( \frac { a \sin b x - b \cos b x } { a ^ { 2 } + b ^ { 2 } } \right)$ + C 
A)  $\mathrm { e } ^ { \mathrm { ax } }$ (sin bx - cos bx) + C 
B)  $\mathrm { e } ^ { \mathrm { ax } }$ (sin bx + cos bx) + C 
C)  $\frac { 1 } { 2 b }$ $\mathrm { e } ^ { \mathrm { ax } }$ $\cos ^ { 2 }$ bx + C 
D)  $\mathrm { e } ^ { \mathrm { ax } }$ $\left( \frac { a \sin b x - b \cos b x } { a ^ { 2 } + b ^ { 2 } } \right)$ + C

Question 11

Evaluate the integral using integration by parts.
-$\int x e ^ { 3 x } d x$

Accepted Answer

A)  $\frac { x ^ { 2 } e ^ { 3 x } } { 6 }$ + C 
B)  $\frac { e ^ { 3 x } } { 3 }$ + C 
C)  $\frac { 3 x \mathrm { e } ^ { 3 x } - \mathrm { e } ^ { 3 x } } { 9 }$ + C 
D)  $\frac { x e ^ { 3 x } - e ^ { 3 x } } { 3 }$ + C 
E)  none of these 
A)  $\frac { x ^ { 2 } e ^ { 3 x } } { 6 }$ + C 
B)  $\frac { e ^ { 3 x } } { 3 }$ + C 
C)  $\frac { 3 x \mathrm { e } ^ { 3 x } - \mathrm { e } ^ { 3 x } } { 9 }$ + C 
D)  $\frac { x e ^ { 3 x } - e ^ { 3 x } } { 3 }$ + C 
E)  none of these

Question 12

A company estimates that the rate of revenue produced by an investment will be K(t) thousand dollars per year at time t, where $\mathrm { K } ( \mathrm { t } ) = 9 \mathrm { t } \mathrm { e } ^ { - 0.2 \mathrm { t } }$ Find the present value of this stream of income over the next four years using 10% interest rate.
Enter just an integer (no units) representing the amount to the nearest dollar.

Accepted Answer

The answer of A company estimates that the rate of...

Question 13

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 } }$

Accepted Answer

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

Question 14

Approximate the integral by the trapezoidal rule. 
-$\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x }$ dx; n = 4 Express your answer to four decimal places.

Accepted Answer

A)  1.2392 
B)  0.6196 
C)  4.9568 
D)  0.9593 
A)  1.2392 
B)  0.6196 
C)  4.9568 
D)  0.9593

Question 15

Evaluate the integral using integration by parts.
-$\int x ^ { 2 }$ $\mathrm { e } ^ { 2 x }$ dx

Accepted Answer

A)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - x } { 2 } \right)$ + 1 + C 
B)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - 4 x } { 2 } \right)$ + C 
C)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - 2 x } { 2 } \right)$ + C 
D)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { 2 x ^ { 2 } - 2 x + 1 } { 4 } \right)$ + C 
A)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - x } { 2 } \right)$ + 1 + C 
B)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - 4 x } { 2 } \right)$ + C 
C)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - 2 x } { 2 } \right)$ + C 
D)  $\mathrm { e } ^ { 2 x }$ $\left( \frac { 2 x ^ { 2 } - 2 x + 1 } { 4 } \right)$ + C

Question 16

Approximate $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx; n = 6, by (a) Simpson's rule and (b) the trapezoidal rule.
Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

Accepted Answer

The answer of Approximate \(\int _ { 1 } ^...

Question 17

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

Accepted Answer

A)  $x ^ { 5 }$ $\left( \frac { 1 } { 5 } \ln x - \frac { 1 } { 5 } \right)$ + C 
B)  $x ^ { 5 }$ $\left( \frac { 1 } { 5 } \ln x - \frac { 1 } { 25 } \right)$ + C 
C)  $x ^ { 5 }$ $\frac { 1 } { 5 } \ln x - 1$ + C 
D)  $x ^ { 5 }$ $\left( 5 \ln x - \frac { 1 } { 25 } \right)$ + C 
A)  $x ^ { 5 }$ $\left( \frac { 1 } { 5 } \ln x - \frac { 1 } { 5 } \right)$ + C 
B)  $x ^ { 5 }$ $\left( \frac { 1 } { 5 } \ln x - \frac { 1 } { 25 } \right)$ + C 
C)  $x ^ { 5 }$ $\frac { 1 } { 5 } \ln x - 1$ + C 
D)  $x ^ { 5 }$ $\left( 5 \ln x - \frac { 1 } { 25 } \right)$ + C

Question 18

$\int \tan ^ { 4 } 2 x \sec ^ { 2 } 2 x d x$ 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 \tan ^ { 4 } 2...

Question 19

Does this integral $\int 3 x \cos x d x$ = 3x sin x + 3 cos x + C?

Accepted Answer

A) True 
 B)False

Question 20

Does this integral $\int 3 x \sqrt { 4 x + 3 } d x$ = $\frac { 3 ( 2 x - 1 ) ( 4 x + 3 ) ^ { 3 / 2 } } { 20 }$ + C?

Accepted Answer

A) True 
 B)False

Determine the integral by making an appropriate substitution. - $\int x ^ { 2 } \sin \left( 7 x ^ { 3 } - 6 \right) d x$

The shaded area in the diagram represents an estimation $\int _ { a } ^ { b } f ( x )$ dx using: $The shaded area in the diagram represents an estimation \int _ { a } ^ { b } f ( x ) dx using:$

Does this integral $\int x \cos 5 x d x$ = $\frac { 1 } { 5 }$ x sin 5x - $\frac { 1 } { 25 }$ cos 5x + C?

Approximate $\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } }$ dx; n = 4, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

Does this integral $\int$ $\frac { \ln ( \ln x ) } { x }$ dx = ln x(ln(ln x) -1) + C?

Determine the integral by making an appropriate substitution. - $\int \frac { \sin \theta \cos ^ { 2 } \theta } { 1 + \cos ^ { 3 } \theta }$ dθ

Determine the integral by making an appropriate substitution. - $\int \frac { \sin x } { 1 - \cos x } d x$

$\int ( x - 1 ) e ^ { \left( 3 x ^ { 2 } - 6 x \right) } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral $\int$ $\frac { \cos x } { ( 3 \sin x + 1 ) ^ { 2 } }$ dx = $\frac { 1 } { 3 ( 3 \sin x + 1 ) }$ + C?

Evaluate the integral using integration by parts. - $\int e ^ { a x } \sin b x d x$

Evaluate the integral using integration by parts. - $\int x e ^ { 3 x } d x$

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 } }$

Approximate the integral by the trapezoidal rule. - $\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x }$ dx; n = 4 Express your answer to four decimal places.

Evaluate the integral using integration by parts. - $\int x ^ { 2 }$ $\mathrm { e } ^ { 2 x }$ dx

Approximate $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx; n = 6, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

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

$\int \tan ^ { 4 } 2 x \sec ^ { 2 } 2 x d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

Does this integral $\int 3 x \cos x d x$ = 3x sin x + 3 cos x + C?

Does this integral $\int 3 x \sqrt { 4 x + 3 } d x$ = $\frac { 3 ( 2 x - 1 ) ( 4 x + 3 ) ^ { 3 / 2 } } { 20 }$ + 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. - ∫x2sin⁡(7x3−6)dx\int x ^ { 2 } \sin \left( 7 x ^ { 3 } - 6 \right) d x∫x2sin(7x3−6)dx

The shaded area in the diagram represents an estimation ∫abf(x)\int _ { a } ^ { b } f ( x )∫ab​f(x) dx using:

Does this integral ∫xcos⁡5xdx\int x \cos 5 x d x∫xcos5xdx = 15\frac { 1 } { 5 }51​ x sin 5x - 125\frac { 1 } { 25 }251​ cos 5x + C?

Approximate ∫0111+x2\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } }∫01​1+x21​ dx; n = 4, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

Does this integral ∫\int∫ ln⁡(ln⁡x)x\frac { \ln ( \ln x ) } { x }xln(lnx)​ dx = ln x(ln(ln x) -1) + C?

Determine the integral by making an appropriate substitution. - ∫sin⁡θcos⁡2θ1+cos⁡3θ\int \frac { \sin \theta \cos ^ { 2 } \theta } { 1 + \cos ^ { 3 } \theta }∫1+cos3θsinθcos2θ​ dθ

Determine the integral by making an appropriate substitution. - ∫sin⁡x1−cos⁡xdx\int \frac { \sin x } { 1 - \cos x } d x∫1−cosxsinx​dx

∫(x−1)e(3x2−6x)dx\int ( x - 1 ) e ^ { \left( 3 x ^ { 2 } - 6 x \right) } d x∫(x−1)e(3x2−6x)dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral ∫\int∫ cos⁡x(3sin⁡x+1)2\frac { \cos x } { ( 3 \sin x + 1 ) ^ { 2 } }(3sinx+1)2cosx​ dx = 13(3sin⁡x+1)\frac { 1 } { 3 ( 3 \sin x + 1 ) }3(3sinx+1)1​ + C?

Evaluate the integral using integration by parts. - ∫eaxsin⁡bxdx\int e ^ { a x } \sin b x d x∫eaxsinbxdx

Evaluate the integral using integration by parts. - ∫xe3xdx\int x e ^ { 3 x } d x∫xe3xdx

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - ∫1∞dxx2\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 } }∫1∞​x2dx​

Approximate the integral by the trapezoidal rule. - ∫01x3ex\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x }∫01​x3ex dx; n = 4 Express your answer to four decimal places.

Evaluate the integral using integration by parts. - ∫x2\int x ^ { 2 }∫x2 e2x\mathrm { e } ^ { 2 x }e2x dx

Approximate ∫121x2\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } }∫12​x21​ dx; n = 6, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

Evaluate the integral using integration by parts. - ∫x4ln⁡xdx\int x ^ { 4 } \ln x d x∫x4lnxdx

∫tan⁡42xsec⁡22xdx\int \tan ^ { 4 } 2 x \sec ^ { 2 } 2 x d x∫tan42xsec22xdx 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.

Does this integral ∫3xcos⁡xdx\int 3 x \cos x d x∫3xcosxdx = 3x sin x + 3 cos x + C?

Does this integral ∫3x4x+3dx\int 3 x \sqrt { 4 x + 3 } d x∫3x4x+3​dx = 3(2x−1)(4x+3)3/220\frac { 3 ( 2 x - 1 ) ( 4 x + 3 ) ^ { 3 / 2 } } { 20 }203(2x−1)(4x+3)3/2​ + 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. - $\int x ^ { 2 } \sin \left( 7 x ^ { 3 } - 6 \right) d x$

The shaded area in the diagram represents an estimation $\int _ { a } ^ { b } f ( x )$ dx using: $The shaded area in the diagram represents an estimation \int _ { a } ^ { b } f ( x ) dx using:$

Does this integral $\int x \cos 5 x d x$ = $\frac { 1 } { 5 }$ x sin 5x - $\frac { 1 } { 25 }$ cos 5x + C?

Approximate $\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } }$ dx; n = 4, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

Does this integral $\int$ $\frac { \ln ( \ln x ) } { x }$ dx = ln x(ln(ln x) -1) + C?

Determine the integral by making an appropriate substitution. - $\int \frac { \sin \theta \cos ^ { 2 } \theta } { 1 + \cos ^ { 3 } \theta }$ dθ

Determine the integral by making an appropriate substitution. - $\int \frac { \sin x } { 1 - \cos x } d x$

$\int ( x - 1 ) e ^ { \left( 3 x ^ { 2 } - 6 x \right) } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral $\int$ $\frac { \cos x } { ( 3 \sin x + 1 ) ^ { 2 } }$ dx = $\frac { 1 } { 3 ( 3 \sin x + 1 ) }$ + C?

Evaluate the integral using integration by parts. - $\int e ^ { a x } \sin b x d x$

Evaluate the integral using integration by parts. - $\int x e ^ { 3 x } d x$

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 2 } }$

Approximate the integral by the trapezoidal rule. - $\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x }$ dx; n = 4 Express your answer to four decimal places.

Evaluate the integral using integration by parts. - $\int x ^ { 2 }$ $\mathrm { e } ^ { 2 x }$ dx

Approximate $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } }$ dx; n = 6, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

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

$\int \tan ^ { 4 } 2 x \sec ^ { 2 } 2 x d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

Does this integral $\int 3 x \cos x d x$ = 3x sin x + 3 cos x + C?

Does this integral $\int 3 x \sqrt { 4 x + 3 } d x$ = $\frac { 3 ( 2 x - 1 ) ( 4 x + 3 ) ^ { 3 / 2 } } { 20 }$ + C?