Question 1

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$.
$$f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$$

Accepted Answer

The answer of Find the average value of the function...

Question 2

The region $\left\{ ( x + y ) \mid x \geq - 6,0 \leq y \leq e ^ { - x / 4 } \right\}$ is represented below.
Find the area of this region to two decimal places. Select the correct answer.

Accepted Answer

A)  $17.89$ 
B)  $15.87$ 
C)  $16.08$ 
D)  $15.89$ 
E)  $17.93$ 
A)  $17.89$ 
B)  $15.87$ 
C)  $16.08$ 
D)  $15.89$ 
E)  $17.93$

Question 3

Evaluate the integral.
$$\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 1 }...

Question 4

Determine whether the integral converges or diverges. If it converges, find its value.
$$\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }$$

Accepted Answer

The answer of Determine whether the integral converges or diverges....

Question 5

Evaluate the integral.
$$\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int \frac { 1 +...

Question 6

Evaluate the integral to six decimal places.
$$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$$

Accepted Answer

The answer of Evaluate the integral to six decimal places.
\[\int...

Question 7

Find the integral. Select the correct answer.
$\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Accepted Answer

A)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 2 } { x - 1 } + C$ 
B)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 1 } { x - 1 } + C$ 
C)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 1 } { x - 1 } + C$ 
D)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 2 } { x - 1 } + C$ 
A)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 2 } { x - 1 } + C$ 
B)  $\ln \left| \frac { x ^ { 2 } } { x - 1 } \right| - \frac { 1 } { x - 1 } + C$ 
C)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 1 } { x - 1 } + C$ 
D)  $\ln \left| \frac { x - 1 } { x ^ { 2 } } \right| - \frac { 2 } { x - 1 } + C$

Question 8

Use long division to evaluate the integral.
$$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$$
The choices are rounded to 3 decimal places.

Accepted Answer

A)  $- 3.544$ 
B)  $6.456$ 
C)  $1.456$ 
D)  $- 8.544$ 
E)  $11.456$ 
A)  $- 3.544$ 
B)  $6.456$ 
C)  $1.456$ 
D)  $- 8.544$ 
E)  $11.456$

Question 9

$$\text { Find the area bounded by the curves } y = 3 \cos x \text { and } y = 3 \cos ^ { 2 } x \text { between } x = 0 \text { and } x = \frac { \pi } { 2 } \text {. }$$

Accepted Answer

The answer of \[\text { Find the area bounded by...

Question 10

Find the integral. Select the correct answer.
$$\int \tan ^ { 2 } x \sec ^ { 6 } x d x$$

Accepted Answer

A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 3 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
C)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x - \tan x + C$ 
D)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x - \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 3 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
C)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x - \tan x + C$ 
D)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x - \frac { 1 } { 3 } \tan ^ { 3 } x + C$

Question 11

Evaluate the integral.
$$\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 0 }...

Question 12

Evaluate the integral using the indicated trigonometric substitution.
$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

Accepted Answer

A)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 5 \sqrt { x ^ { 2 } + 25 } + C$ 
B)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 25 \sqrt { x ^ { 2 } + 25 } + C$ 
E)  $\frac { 3 } { 2 } ( x + 25 ) ^ { 3 / 2 } - 25 \sqrt { x + 25 } + C$ 
A)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 5 \sqrt { x ^ { 2 } + 25 } + C$ 
B)  $\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 25 \sqrt { x ^ { 2 } + 25 } + C$ 
E)  $\frac { 3 } { 2 } ( x + 25 ) ^ { 3 / 2 } - 25 \sqrt { x + 25 } + C$

Question 13

Use the Table of Integrals to evaluate the integral.
$$\int e ^ { 8 x } \sin 4 x d x$$

Accepted Answer

The answer of Use the Table of Integrals to evaluate...

Question 14

Find the integral.
$$\int x e ^ { 3 x } d x$$

Accepted Answer

The answer of Find the integral.
\[\int x e ^ {...

Question 15

A manufacturer of light bulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let $F ( t )$ be the fraction of the company's bulbs that burn out before $t$ hours. $F ( t )$ lies between 0 and 1 .
Let $r ( t ) = F ^ { \prime } ( t )$. What is the value of $\int _ { 0 } ^ { \infty } r ( t ) d t$ ?

Accepted Answer

The answer of A manufacturer of light bulbs wants to...

Question 16

Find the integral.
$$\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x$$

Accepted Answer

The answer of Find the integral.
\[\int \frac { 3 x...

Question 17

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places.
$$\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$$

Accepted Answer

A)  $2.0076$ 
B)  $2.9504$ 
C)  $2.6098$ 
D)  $2.2955$ 
A)  $2.0076$ 
B)  $2.9504$ 
C)  $2.6098$ 
D)  $2.2955$

Question 18

Evaluate the integral.
$$\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int \frac { 6 \sin...

Question 19

Find a bound on the error in approximating the integral $\int _ { 1 } ^ { 9 } \ln x ^ { 9 }$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 10$ subintervals.

Accepted Answer

The answer of Find a bound on the error in...

Question 20

Evaluate the integral.
$$\int 15 x \ln ( 1 + x ) d x$$

Accepted Answer

The answer of Evaluate the integral.
\[\int 15 x \ln (...

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

Determine whether the integral converges or diverges. If it converges, find its value. $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }$

Evaluate the integral. $\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x$

Evaluate the integral to six decimal places. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$

Find the integral. Select the correct answer. $\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

$\text { Find the area bounded by the curves } y = 3 \cos x \text { and } y = 3 \cos ^ { 2 } x \text { between } x = 0 \text { and } x = \frac { \pi } { 2 } \text {. }$

Find the integral. Select the correct answer. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

Use the Table of Integrals to evaluate the integral. $\int e ^ { 8 x } \sin 4 x d x$

Find the integral. $\int x e ^ { 3 x } d x$

Find the integral. $\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Find a bound on the error in approximating the integral $\int _ { 1 } ^ { 9 } \ln x ^ { 9 }$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 10$ subintervals.

Evaluate the integral. $\int 15 x \ln ( 1 + x ) d x$

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Exam 7: Techniques of Integration

Find the average value of the function f(x)f ( x )f(x) in the interval [−π,π][ - \pi , \pi ][−π,π] . f(x)=sin⁡6xcos⁡5xf ( x ) = \sin ^ { 6 } x \cos ^ { 5 } xf(x)=sin6xcos5x

The region {(x+y)∣x≥−6,0≤y≤e−x/4}\left\{ ( x + y ) \mid x \geq - 6,0 \leq y \leq e ^ { - x / 4 } \right\}{(x+y)∣x≥−6,0≤y≤e−x/4} is represented below. Find the area of this region to two decimal places. Select the correct answer.

Evaluate the integral. ∫1∞dxxln⁡x\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }∫1∞​xlnxdx​

Determine whether the integral converges or diverges. If it converges, find its value. ∫1∞dxx5ln⁡x\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }∫1∞​x5lnxdx​

Evaluate the integral. ∫1+3ex1−exdx\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x∫1−ex1+3ex​dx

Evaluate the integral to six decimal places. ∫01x364−x2dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x∫01​64−x2​x3​dx

Find the integral. Select the correct answer. ∫x2−x+2x3−2x2+xdx\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x∫x3−2x2+xx2−x+2​dx

Use long division to evaluate the integral. ∫01x3+12x2−36x+1x2+4x−12dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x∫01​x2+4x−12x3+12x2−36x+1​dx The choices are rounded to 3 decimal places.

Find the integral. Select the correct answer. ∫tan⁡2xsec⁡6xdx\int \tan ^ { 2 } x \sec ^ { 6 } x d x∫tan2xsec6xdx

Evaluate the integral. ∫01/212xcos⁡πxdx\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x∫01/2​12xcosπxdx

Evaluate the integral using the indicated trigonometric substitution. ∫x3x2+25dx;x=5tan⁡θ\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta∫x2+25​x3​dx;x=5tanθ

Use the Table of Integrals to evaluate the integral. ∫e8xsin⁡4xdx\int e ^ { 8 x } \sin 4 x d x∫e8xsin4xdx

Find the integral. ∫xe3xdx\int x e ^ { 3 x } d x∫xe3xdx

Find the integral. ∫3x2−7x−2(x2+x+1)2dx\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x∫(x2+x+1)23x2−7x−2​dx

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. ∫−11x2+1dx;n=6\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6∫−11​x2+1​dx;n=6

Evaluate the integral. ∫6sin⁡2x1+cos⁡4xdx\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x∫1+cos4x6sin2x​dx

Find a bound on the error in approximating the integral ∫19ln⁡x9\int _ { 1 } ^ { 9 } \ln x ^ { 9 }∫19​lnx9 using (a) the Trapezoidal Rule and (b) Simpson's Rule with n=10n = 10n=10 subintervals.

Evaluate the integral. ∫15xln⁡(1+x)dx\int 15 x \ln ( 1 + x ) d x∫15xln(1+x)dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

Determine whether the integral converges or diverges. If it converges, find its value. $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 5 } \ln x }$

Evaluate the integral. $\int \frac { 1 + 3 e ^ { x } } { 1 - e ^ { x } } d x$

Evaluate the integral to six decimal places. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { \sqrt { 64 - x ^ { 2 } } } d x$

Find the integral. Select the correct answer. $\int \frac { x ^ { 2 } - x + 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Find the integral. Select the correct answer. $\int \tan ^ { 2 } x \sec ^ { 6 } x d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 / 2 } 12 x \cos \pi x d x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

Use the Table of Integrals to evaluate the integral. $\int e ^ { 8 x } \sin 4 x d x$

Find the integral. $\int x e ^ { 3 x } d x$

Find the integral. $\int \frac { 3 x ^ { 2 } - 7 x - 2 } { \left( x ^ { 2 } + x + 1 \right) ^ { 2 } } d x$

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places. $\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } + 1 } d x ; \quad n = 6$

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Find a bound on the error in approximating the integral $\int _ { 1 } ^ { 9 } \ln x ^ { 9 }$ using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 10$ subintervals.

Evaluate the integral. $\int 15 x \ln ( 1 + x ) d x$