Question 1

$\int 4 x ^ { 5 } \left( x ^ { 6 } + 100 \right) ^ { 5 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Accepted Answer

The answer of \(\int 4 x ^ { 5 }...

Question 2

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

Accepted Answer

A)  π 
B)  -1 
C)  0 
D)  1 
A)  π 
B)  -1 
C)  0 
D)  1

Question 3

Approximate $\int _ { 1 } ^ { 5 } \frac { 1 } { 2 x ^ { 2 } } d x$ ; n = 8, by (a) the trapezoidal rule, (b) the midpoint rule, and (c) then find the exact value of the integral.
Enter just a, b, c as real numbers all rounded to two decimal places. Do not label, but answer in the above order using commas to separate.

Accepted Answer

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

Question 4

Approximate $\int _ { 0 } ^ { 1 } x ^ { 4 }$ 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

$\int$ $\frac { \cos 2 x } { \sin ^ { 3 } 2 x }$ 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 $\int$ \(\frac { \cos 2 x }...

Question 6

$\int _ { - \infty } ^ { 0 } e ^ { 3 x + 1 } d x$ Enter your answer as a reduced quotient or the word "divergent".

Accepted Answer

The answer of \(\int _ { - \infty } ^...

Question 7

Does this integral $\int _ { 1 } ^ { \sqrt { \mathrm { e } } } \frac { \cos \left( \ln x ^ { 2 } \right) } { x }$ dx = $\frac { 1 } { 2 }$ sin(1)?

Accepted Answer

A) True 
 B)False

Question 8

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

Accepted Answer

The answer of \(\int \left( 3 x ^ { 2...

Question 9

Does this integral $\int \sin ( \ln x ) d x$ = $\frac { x \sin ( \ln x ) - x \cos ( \ln x ) } { 2 }$ ?

Accepted Answer

A) True 
 B)False

Question 10

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { - \infty } ^ { 0 } e ^ { 10 x }$ dx

Accepted Answer

A)  0 
B)  -1 
C)  1 
D)  divergent 
E)  $\frac { 1 } { 10 }$ 
A)  0 
B)  -1 
C)  1 
D)  divergent 
E)  $\frac { 1 } { 10 }$

Question 11

Determine the integral by making an appropriate substitution. 
-$\int 4 ( 2 x + 5 ) ^ { 3 } d x$

Accepted Answer

A)  $\frac { 1 } { 4 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
B)  $\frac { 3 } { 8 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
C)  $\frac { 3 } { 4 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
D)  $\frac { 1 } { 2 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
A)  $\frac { 1 } { 4 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
B)  $\frac { 3 } { 8 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
C)  $\frac { 3 } { 4 }$ $( 2 x + 5 ) ^ { 4 }$ + C 
D)  $\frac { 1 } { 2 }$ $( 2 x + 5 ) ^ { 4 }$ + C

Question 12

Find the area of the shaded region.
y = $\frac { 4 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$

Accepted Answer

A)  - 1 square units 
B)  1/3 square units 
C)  5/3 square units 
D)  2 square units 
A)  - 1 square units 
B)  1/3 square units 
C)  5/3 square units 
D)  2 square units

Question 13

Evaluate the integral using integration by parts.
-$\int \sin ^ { - 1 } x$ dx

Accepted Answer

A)  $\sin ^ { - 1 }$ x + $\sqrt { 1 - x ^ { 2 } }$ + C 
B)  x + $\sqrt { 1 - x ^ { 2 } }$ + C 
C)  x $\sin ^ { - 1 }$ x + $\sqrt { 1 - x ^ { 2 } }$ + C 
D)  $\sin ^ { - 1 }$ x - $\sqrt { 1 - x ^ { 2 } }$ + C 
A)  $\sin ^ { - 1 }$ x + $\sqrt { 1 - x ^ { 2 } }$ + C 
B)  x + $\sqrt { 1 - x ^ { 2 } }$ + C 
C)  x $\sin ^ { - 1 }$ x + $\sqrt { 1 - x ^ { 2 } }$ + C 
D)  $\sin ^ { - 1 }$ x - $\sqrt { 1 - x ^ { 2 } }$ + C

Question 14

Does this integral $\int$ $\frac { x + 1 } { e ^ { 2 x } }$ dx = $\frac { - \mathrm { e } ^ { 2 \mathrm { x } } } { 4 }$ (2x + 3) + C?

Accepted Answer

A) True 
 B)False

Question 15

If the annual rate of return from an investment is -3000 + 125t, find the present value of the income generated in the third year if the interest rates are 8.5%.

Accepted Answer

A)  $\int _ { 0 } ^ { 3 } ( - 3000 + 125 t ) ^ { - 0.085 t } d t$ 
B)  $\int _ { 2 } ^ { 3 } ( 125 t - 3000 ) e ^ { 0.085 t } d t$ 
C)  $\int _ { 0 } ^ { 3 } ( 125 t - 3000 ) e ^ { 0.085 t } d t$ 
D)  $\int _ { 2 } ^ { 3 } ( - 3000 + 125 t ) ^ { - 0.085 t } d t$ 
E)  none of these 
A)  $\int _ { 0 } ^ { 3 } ( - 3000 + 125 t ) ^ { - 0.085 t } d t$ 
B)  $\int _ { 2 } ^ { 3 } ( 125 t - 3000 ) e ^ { 0.085 t } d t$ 
C)  $\int _ { 0 } ^ { 3 } ( 125 t - 3000 ) e ^ { 0.085 t } d t$ 
D)  $\int _ { 2 } ^ { 3 } ( - 3000 + 125 t ) ^ { - 0.085 t } d t$ 
E)  none of these

Question 16

Determine the integral by making an appropriate substitution. 
-$\int \cos ^ { 1 / 3 } x \sin x$ dx

Accepted Answer

A)  $\cos 4 / 3$ x sin x + C 
B)  $\sin ^ { 2 }$ x + C 
C)  - $\frac { 3 } { 4 }$ $\cos 4 / 3$ x + C 
D)  $\cos 1 / 3$ x + C 
E)  none of these 
A)  $\cos 4 / 3$ x sin x + C 
B)  $\sin ^ { 2 }$ x + C 
C)  - $\frac { 3 } { 4 }$ $\cos 4 / 3$ x + C 
D)  $\cos 1 / 3$ x + C 
E)  none of these

Question 17

Approximate $\int _ { 0 } ^ { 1 } x ^ { 3 }$ 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 18

Consider $\lim _ { b \rightarrow \infty }$ $\frac { 2 b - 1 } { b }$ . Which of the following is true?

Accepted Answer

A)  The limit exists and is equal to zero. 
B)  none of these 
C)  The limit exists and is equal to one. 
D)  The limit diverges. 
E)  The limit exists and is equal to two.
Evaluate the improper integral whenever it is convergent. If it is divergent, state this. 
A)  The limit exists and is equal to zero. 
B)  none of these 
C)  The limit exists and is equal to one. 
D)  The limit diverges. 
E)  The limit exists and is equal to two.
Evaluate the improper integral whenever it is convergent. If it is divergent, state this.

Question 19

$\int$ $\frac { \sin \sqrt { x } } { \sqrt { x } }$ dx
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 { \sin \sqrt { x...

Question 20

$\int _ { 0 } ^ { \infty } x ( x + 1 ) ^ { - 3 } d x$ Enter your answer as a reduced fraction or the word "divergent".

Accepted Answer

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

$\int 4 x ^ { 5 } \left( x ^ { 6 } + 100 \right) ^ { 5 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

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

Approximate $\int _ { 0 } ^ { 1 } x ^ { 4 }$ 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.

$\int$ $\frac { \cos 2 x } { \sin ^ { 3 } 2 x }$ 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.

$\int _ { - \infty } ^ { 0 } e ^ { 3 x + 1 } d x$ Enter your answer as a reduced quotient or the word "divergent".

Does this integral $\int _ { 1 } ^ { \sqrt { \mathrm { e } } } \frac { \cos \left( \ln x ^ { 2 } \right) } { x }$ dx = $\frac { 1 } { 2 }$ sin(1)?

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

Does this integral $\int \sin ( \ln x ) d x$ = $\frac { x \sin ( \ln x ) - x \cos ( \ln x ) } { 2 }$ ?

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { - \infty } ^ { 0 } e ^ { 10 x }$ dx

Determine the integral by making an appropriate substitution. - $\int 4 ( 2 x + 5 ) ^ { 3 } d x$

Find the area of the shaded region. y = $\frac { 4 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ $Find the area of the shaded region. y = \frac { 4 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$

Evaluate the integral using integration by parts. - $\int \sin ^ { - 1 } x$ dx

Does this integral $\int$ $\frac { x + 1 } { e ^ { 2 x } }$ dx = $\frac { - \mathrm { e } ^ { 2 \mathrm { x } } } { 4 }$ (2x + 3) + C?

If the annual rate of return from an investment is -3000 + 125t, find the present value of the income generated in the third year if the interest rates are 8.5%.

Determine the integral by making an appropriate substitution. - $\int \cos ^ { 1 / 3 } x \sin x$ dx

Approximate $\int _ { 0 } ^ { 1 } x ^ { 3 }$ 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.

Consider $\lim _ { b \rightarrow \infty }$ $\frac { 2 b - 1 } { b }$ . Which of the following is true?

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

$\int _ { 0 } ^ { \infty } x ( x + 1 ) ^ { - 3 } d x$ Enter your answer as a reduced fraction or the word "divergent".

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

∫4x5(x6+100)5dx\int 4 x ^ { 5 } \left( x ^ { 6 } + 100 \right) ^ { 5 } d x∫4x5(x6+100)5dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Evaluate the integral. - ∫0πsin⁡xcos⁡xdx\int _ { 0 } ^ { \pi } \sin x \cos x d x∫0π​sinxcosxdx

Approximate ∫01x4\int _ { 0 } ^ { 1 } x ^ { 4 }∫01​x4 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.

∫\int∫ cos⁡2xsin⁡32x\frac { \cos 2 x } { \sin ^ { 3 } 2 x }sin32xcos2x​ dx 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.

∫−∞0e3x+1dx\int _ { - \infty } ^ { 0 } e ^ { 3 x + 1 } d x∫−∞0​e3x+1dx Enter your answer as a reduced quotient or the word "divergent".

Does this integral ∫1ecos⁡(ln⁡x2)x\int _ { 1 } ^ { \sqrt { \mathrm { e } } } \frac { \cos \left( \ln x ^ { 2 } \right) } { x }∫1e​​xcos(lnx2)​ dx = 12\frac { 1 } { 2 }21​ sin(1)?

∫(3x2+2)x3+2xdx\int \left( 3 x ^ { 2 } + 2 \right) \sqrt { x ^ { 3 } + 2 x } d x∫(3x2+2)x3+2x​dx Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Does this integral ∫sin⁡(ln⁡x)dx\int \sin ( \ln x ) d x∫sin(lnx)dx = xsin⁡(ln⁡x)−xcos⁡(ln⁡x)2\frac { x \sin ( \ln x ) - x \cos ( \ln x ) } { 2 }2xsin(lnx)−xcos(lnx)​ ?

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - ∫−∞0e10x\int _ { - \infty } ^ { 0 } e ^ { 10 x }∫−∞0​e10x dx

Determine the integral by making an appropriate substitution. - ∫4(2x+5)3dx\int 4 ( 2 x + 5 ) ^ { 3 } d x∫4(2x+5)3dx

Find the area of the shaded region. y = 4x(2x2+1)2\frac { 4 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }(2x2+1)24x​

Evaluate the integral using integration by parts. - ∫sin⁡−1x\int \sin ^ { - 1 } x∫sin−1x dx

Does this integral ∫\int∫ x+1e2x\frac { x + 1 } { e ^ { 2 x } }e2xx+1​ dx = −e2x4\frac { - \mathrm { e } ^ { 2 \mathrm { x } } } { 4 }4−e2x​ (2x + 3) + C?

If the annual rate of return from an investment is -3000 + 125t, find the present value of the income generated in the third year if the interest rates are 8.5%.

Determine the integral by making an appropriate substitution. - ∫cos⁡1/3xsin⁡x\int \cos ^ { 1 / 3 } x \sin x∫cos1/3xsinx dx

Approximate ∫01x3\int _ { 0 } ^ { 1 } x ^ { 3 }∫01​x3 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.

Consider lim⁡b→∞\lim _ { b \rightarrow \infty }limb→∞​ 2b−1b\frac { 2 b - 1 } { b }b2b−1​ . Which of the following is true?

∫\int∫ sin⁡xx\frac { \sin \sqrt { x } } { \sqrt { x } }x​sinx​​ dx Enter your answer with any coefficients in front as integers or reduced fractions of form ab\frac { a } { b }ba​ .

∫0∞x(x+1)−3dx\int _ { 0 } ^ { \infty } x ( x + 1 ) ^ { - 3 } d x∫0∞​x(x+1)−3dx Enter your answer as a reduced fraction or the word "divergent".

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

$\int 4 x ^ { 5 } \left( x ^ { 6 } + 100 \right) ^ { 5 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

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

Approximate $\int _ { 0 } ^ { 1 } x ^ { 4 }$ 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.

$\int$ $\frac { \cos 2 x } { \sin ^ { 3 } 2 x }$ 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.

$\int _ { - \infty } ^ { 0 } e ^ { 3 x + 1 } d x$ Enter your answer as a reduced quotient or the word "divergent".

Does this integral $\int _ { 1 } ^ { \sqrt { \mathrm { e } } } \frac { \cos \left( \ln x ^ { 2 } \right) } { x }$ dx = $\frac { 1 } { 2 }$ sin(1)?

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

Does this integral $\int \sin ( \ln x ) d x$ = $\frac { x \sin ( \ln x ) - x \cos ( \ln x ) } { 2 }$ ?

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { - \infty } ^ { 0 } e ^ { 10 x }$ dx

Determine the integral by making an appropriate substitution. - $\int 4 ( 2 x + 5 ) ^ { 3 } d x$

Find the area of the shaded region. y = $\frac { 4 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ $Find the area of the shaded region. y = \frac { 4 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$

Evaluate the integral using integration by parts. - $\int \sin ^ { - 1 } x$ dx

Does this integral $\int$ $\frac { x + 1 } { e ^ { 2 x } }$ dx = $\frac { - \mathrm { e } ^ { 2 \mathrm { x } } } { 4 }$ (2x + 3) + C?

Determine the integral by making an appropriate substitution. - $\int \cos ^ { 1 / 3 } x \sin x$ dx

Approximate $\int _ { 0 } ^ { 1 } x ^ { 3 }$ 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.

Consider $\lim _ { b \rightarrow \infty }$ $\frac { 2 b - 1 } { b }$ . Which of the following is true?

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

$\int _ { 0 } ^ { \infty } x ( x + 1 ) ^ { - 3 } d x$ Enter your answer as a reduced fraction or the word "divergent".