Question 1

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

Accepted Answer

A) True 
 B)False  False

Question 2

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

Accepted Answer

A)  $\frac { 1 } { 6 }$ $x ^ { 2 }$ $e ^ { 3 x }$ + C 
B)  $e ^ { 3 x }$ $\left( \frac { 1 } { 3 } x - \frac { 1 } { 9 } \right)$ + C 
C)  $e ^ { 3 x }$ $\left( \frac { 1 } { 3 } x - 1 \right)$ + C 
D)  $\frac { 1 } { 3 }$ $e ^ { 3 x }$ + C 
A)  $\frac { 1 } { 6 }$ $x ^ { 2 }$ $e ^ { 3 x }$ + C 
B)  $e ^ { 3 x }$ $\left( \frac { 1 } { 3 } x - \frac { 1 } { 9 } \right)$ + C 
C)  $e ^ { 3 x }$ $\left( \frac { 1 } { 3 } x - 1 \right)$ + C 
D)  $\frac { 1 } { 3 }$ $e ^ { 3 x }$ + C B

Question 3

Does this integral $\int _ { 0 } ^ { 1 }$ $\frac { 2 x ^ { 3 } } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }$ dx = ln 5 - 2 ln 2 - $\frac { 1 } { 5 }$ ?

Accepted Answer

A) True 
 B)False  True

Question 4

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

Accepted Answer

A) True 
 B)False

Question 5

Find the area under the graph of y = $x e ^ { - x ^ { 2 } }$ for x ≥ 0. Enter just a reduced fraction.

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Does this integral $\int x \sin 10 x d x$ = $\frac { 1 } { 100 }$ sin 10x - $\frac { x } { 10 }$ cos 10x + C?

Accepted Answer

A) True 
 B)False

Question 8

Evaluate the integral. 
-$\int _ { 0 } ^ { 1 } 5 x 4 \sqrt { x ^ { 5 } + 4 }$ dx

Accepted Answer

A)  $\frac { 10 \sqrt { 5 } + 16 } { 3 }$ 
B)  $\frac { 10 \sqrt { 5 } - 16 } { 15 }$ 
C)  $\frac { 10 \sqrt { 5 } - 16 } { 3 }$ 
D)  2 $\sqrt { 5 }$ -4 
A)  $\frac { 10 \sqrt { 5 } + 16 } { 3 }$ 
B)  $\frac { 10 \sqrt { 5 } - 16 } { 15 }$ 
C)  $\frac { 10 \sqrt { 5 } - 16 } { 3 }$ 
D)  2 $\sqrt { 5 }$ -4

Question 9

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

Accepted Answer

The answer of \(\int e ^ { x } \sqrt...

Question 10

Suppose the annual rate of income from an investment at any time t is $K ( t ) = - 100 + 50 t$ . What is the formula for the present value of the income over the next 5 years at a 6% interest rate compounded continuously?

Accepted Answer

A)  $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { 0.06 t } d t$ 
B)  $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$ 
C)  $\int _ { 1 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$ 
D)  $\int _ { 1 } ^ { 5 } 50 t ^ { - 0.06 t } d t$ 
E)  none of these 
A)  $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { 0.06 t } d t$ 
B)  $\int _ { 0 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$ 
C)  $\int _ { 1 } ^ { 5 } ( 50 t - 100 ) e ^ { - 0.06 t } d t$ 
D)  $\int _ { 1 } ^ { 5 } 50 t ^ { - 0.06 t } d t$ 
E)  none of these

Question 11

$\int$ $\frac { e ^ { x } + x e ^ { x } } { 4 + x e ^ { x } }$ 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 12

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { 3 } ^ { \infty } e ^ { - x / 2 } d x$

Accepted Answer

A)  divergent 
B)  $\mathrm { e } ^ { - 1 }$ 
C)  2 $e ^ { - 3 / 2 }$ 
D)  $e ^ { - 3 / 2 }$ 
E)  none of these 
A)  divergent 
B)  $\mathrm { e } ^ { - 1 }$ 
C)  2 $e ^ { - 3 / 2 }$ 
D)  $e ^ { - 3 / 2 }$ 
E)  none of these

Question 13

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

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  divergent 
E)  -1 
A)  0 
B)  1 
C)  2 
D)  divergent 
E)  -1

Question 14

$\int \sin ^ { 4 } x \cos x d x$ 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 ^ { 4 } x...

Question 15

Approximate the integral by the trapezoidal rule. 
-$\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$ dx; n = 4 Express your answer to five decimal places.

Accepted Answer

A)  23.42223 
B)  147.19132 
C)  36.79783 
D)  73.59566 
A)  23.42223 
B)  147.19132 
C)  36.79783 
D)  73.59566

Question 16

Evaluate the integral using integration by parts.
-$\int \ln 2 x d x$

Accepted Answer

A)  x(ln|2x| + 1) + C 
B)  x(ln|2x| - 2) + C 
C)  x(ln|2x| - 1) + C 
D)  x(ln|2x| + 2) + C 
A)  x(ln|2x| + 1) + C 
B)  x(ln|2x| - 2) + C 
C)  x(ln|2x| - 1) + C 
D)  x(ln|2x| + 2) + C

Question 17

A homeowner has fences on three sides of her property and a stream runs along the fourth side. She makes measurements of the distance to the stream every 10 feet as illustrated. What is the approximate area of her property?   Enter just an integer (no units).

Accepted Answer

The answer of A homeowner has fences on three sides...

Question 18

Does this integral $\int t ^ { 3 } \cos t d t$ = ( $a ^ { 3 }$ - 6t) sin t + 3( $t ^ { 2 }$ - 2) cos t + C?

Accepted Answer

A) True 
 B)False

Question 19

$\int x \sqrt { x ^ { 2 } - 16 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Accepted Answer

The answer of \(\int x \sqrt { x ^ {...

Question 20

Does this integral $\int x ^ { 2 } \sin 2 x d x$ = $\frac { \left( 1 - 2 x ^ { 2 } \right) \cos 2 x } { 4 }$ + $\frac { x \sin 2 x } { 2 }$ + C?

Accepted Answer

A) True 
 B)False

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

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

Does this integral $\int _ { 0 } ^ { 1 }$ $\frac { 2 x ^ { 3 } } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }$ dx = ln 5 - 2 ln 2 - $\frac { 1 } { 5 }$ ?

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

Find the area under the graph of y = $x e ^ { - x ^ { 2 } }$ for x ≥ 0. Enter just a reduced fraction.

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

Does this integral $\int x \sin 10 x d x$ = $\frac { 1 } { 100 }$ sin 10x - $\frac { x } { 10 }$ cos 10x + C?

Evaluate the integral. - $\int _ { 0 } ^ { 1 } 5 x 4 \sqrt { x ^ { 5 } + 4 }$ dx

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

Suppose the annual rate of income from an investment at any time t is $K ( t ) = - 100 + 50 t$ . What is the formula for the present value of the income over the next 5 years at a 6% interest rate compounded continuously?

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

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 3 } ^ { \infty } e ^ { - x / 2 } d x$

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

$\int \sin ^ { 4 } x \cos x d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Approximate the integral by the trapezoidal rule. - $\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$ dx; n = 4 Express your answer to five decimal places.

Evaluate the integral using integration by parts. - $\int \ln 2 x d x$

A homeowner has fences on three sides of her property and a stream runs along the fourth side. She makes measurements of the distance to the stream every 10 feet as illustrated. What is the approximate area of her property? Enter just an integer (no units).

Does this integral $\int t ^ { 3 } \cos t d t$ = ( $a ^ { 3 }$ - 6t) sin t + 3( $t ^ { 2 }$ - 2) cos t + C?

$\int x \sqrt { x ^ { 2 } - 16 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral $\int x ^ { 2 } \sin 2 x d x$ = $\frac { \left( 1 - 2 x ^ { 2 } \right) \cos 2 x } { 4 }$ + $\frac { x \sin 2 x } { 2 }$ + 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

Exam 9: Techniques of Integration

Does this integral ∫ln⁡(2x)x2dx\int \frac { \ln ( 2 x ) } { x ^ { 2 } } d x∫x2ln(2x)​dx = - 1+ln⁡2+ln⁡xx\frac { 1 + \ln 2 + \ln x } { x }x1+ln2+lnx​ + C?

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

Does this integral ∫01\int _ { 0 } ^ { 1 }∫01​ 2x3(x2+4)2\frac { 2 x ^ { 3 } } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }(x2+4)22x3​ dx = ln 5 - 2 ln 2 - 15\frac { 1 } { 5 }51​ ?

Does this integral ∫2xexdx\int 2 x e ^ { x } d x∫2xexdx = 2 exe ^ { x }ex (x - 1) + C?

Find the area under the graph of y = xe−x2x e ^ { - x ^ { 2 } }xe−x2 for x ≥ 0. Enter just a reduced fraction.

Evaluate the integral. - ∫1412x+1dx\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } d x∫14​2x+11​dx

Does this integral ∫xsin⁡10xdx\int x \sin 10 x d x∫xsin10xdx = 1100\frac { 1 } { 100 }1001​ sin 10x - x10\frac { x } { 10 }10x​ cos 10x + C?

Evaluate the integral. - ∫015x4x5+4\int _ { 0 } ^ { 1 } 5 x 4 \sqrt { x ^ { 5 } + 4 }∫01​5x4x5+4​ dx

∫exex+2dx\int e ^ { x } \sqrt { e ^ { x } + 2 } d x∫exex+2​dx Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ .

Suppose the annual rate of income from an investment at any time t is K(t)=−100+50tK ( t ) = - 100 + 50 tK(t)=−100+50t . What is the formula for the present value of the income over the next 5 years at a 6% interest rate compounded continuously?

∫\int∫ ex+xex4+xex\frac { e ^ { x } + x e ^ { x } } { 4 + x e ^ { x } }4+xexex+xex​ dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - ∫3∞e−x/2dx\int _ { 3 } ^ { \infty } e ^ { - x / 2 } d x∫3∞​e−x/2dx

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - ∫−∞∞xe−x2\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } }∫−∞∞​xe−x2 dx

∫sin⁡4xcos⁡xdx\int \sin ^ { 4 } x \cos x d x∫sin4xcosxdx Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ .

Approximate the integral by the trapezoidal rule. - ∫46(ln⁡x)6\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }∫46​(lnx)6 dx; n = 4 Express your answer to five decimal places.

Evaluate the integral using integration by parts. - ∫ln⁡2xdx\int \ln 2 x d x∫ln2xdx

A homeowner has fences on three sides of her property and a stream runs along the fourth side. She makes measurements of the distance to the stream every 10 feet as illustrated. What is the approximate area of her property? Enter just an integer (no units).

Does this integral ∫t3cos⁡tdt\int t ^ { 3 } \cos t d t∫t3costdt = ( a3a ^ { 3 }a3 - 6t) sin t + 3( t2t ^ { 2 }t2 - 2) cos t + C?

∫xx2−16dx\int x \sqrt { x ^ { 2 } - 16 } d x∫xx2−16​dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral ∫x2sin⁡2xdx\int x ^ { 2 } \sin 2 x d x∫x2sin2xdx = (1−2x2)cos⁡2x4\frac { \left( 1 - 2 x ^ { 2 } \right) \cos 2 x } { 4 }4(1−2x2)cos2x​ + xsin⁡2x2\frac { x \sin 2 x } { 2 }2xsin2x​ + 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

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

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

Does this integral $\int _ { 0 } ^ { 1 }$ $\frac { 2 x ^ { 3 } } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }$ dx = ln 5 - 2 ln 2 - $\frac { 1 } { 5 }$ ?

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

Find the area under the graph of y = $x e ^ { - x ^ { 2 } }$ for x ≥ 0. Enter just a reduced fraction.

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

Does this integral $\int x \sin 10 x d x$ = $\frac { 1 } { 100 }$ sin 10x - $\frac { x } { 10 }$ cos 10x + C?

Evaluate the integral. - $\int _ { 0 } ^ { 1 } 5 x 4 \sqrt { x ^ { 5 } + 4 }$ dx

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

Suppose the annual rate of income from an investment at any time t is $K ( t ) = - 100 + 50 t$ . What is the formula for the present value of the income over the next 5 years at a 6% interest rate compounded continuously?

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

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 3 } ^ { \infty } e ^ { - x / 2 } d x$

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

$\int \sin ^ { 4 } x \cos x d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Approximate the integral by the trapezoidal rule. - $\int _ { 4 } ^ { 6 } ( \ln x ) ^ { 6 }$ dx; n = 4 Express your answer to five decimal places.

Evaluate the integral using integration by parts. - $\int \ln 2 x d x$

Does this integral $\int t ^ { 3 } \cos t d t$ = ( $a ^ { 3 }$ - 6t) sin t + 3( $t ^ { 2 }$ - 2) cos t + C?

$\int x \sqrt { x ^ { 2 } - 16 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral $\int x ^ { 2 } \sin 2 x d x$ = $\frac { \left( 1 - 2 x ^ { 2 } \right) \cos 2 x } { 4 }$ + $\frac { x \sin 2 x } { 2 }$ + C?