Question 1

Find the integral.
$\int \frac { d x } { x ( x - 3 ) }$

Accepted Answer

A)  $\frac { 1 } { 3 } \ln \left| \frac { x - 3 } { x } \right| + C$ 
B)  $\frac { 1 } { 3 } \ln \left| \frac { x ^ { 2 } - 3 } { x } \right| + C$ 
C)  $3 \ln \left| \frac { x ^ { 2 } - 3 } { x } \right| + C$ 
D)  $3 \ln \left| \frac { x - 3 } { x } \right| + C$ 
A)  $\frac { 1 } { 3 } \ln \left| \frac { x - 3 } { x } \right| + C$ 
B)  $\frac { 1 } { 3 } \ln \left| \frac { x ^ { 2 } - 3 } { x } \right| + C$ 
C)  $3 \ln \left| \frac { x ^ { 2 } - 3 } { x } \right| + C$ 
D)  $3 \ln \left| \frac { x - 3 } { x } \right| + C$

Question 2

Evaluate the integral or show that it is divergent. $$\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$$

Accepted Answer

A)  $- \frac { \pi } { 10 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $\frac { \pi } { 8 }$ 
D)  $\frac { 5 } { 4 } \pi$ 
E)  divergent 
A)  $- \frac { \pi } { 10 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $\frac { \pi } { 8 }$ 
D)  $\frac { 5 } { 4 } \pi$ 
E)  divergent

Question 3

Evaluate the integral. $$\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$$

Accepted Answer

A)  $- \frac { 1 } { 5 } ( \ln 2 ) ^ { 2 }$ 
B)  $- \frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$ 
C)  0 
D)  $\frac { 1 } { 6 } ( \ln 3 ) ^ { 2 }$ 
E)  $\frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$ 
A)  $- \frac { 1 } { 5 } ( \ln 2 ) ^ { 2 }$ 
B)  $- \frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$ 
C)  0 
D)  $\frac { 1 } { 6 } ( \ln 3 ) ^ { 2 }$ 
E)  $\frac { 1 } { 2 } ( \ln 3 ) ^ { 2 }$

Question 4

Determine whether the improper integral converges or diverges, and if it converges, find its value. $$\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$$

Accepted Answer

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

Question 5

Evaluate the integral. $$\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x$$

Accepted Answer

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

Question 6

Let $a$ and $b$ be real numbers. What integral must appear in place of the question mark "?" to make the following statement true?
$$\int _ { - \infty } ^ { a } \frac { 10 } { x ^ { 2 } + 9 } d x + \int _ { a } ^ { \infty } \frac { 10 } { x ^ { 2 } + 9 } d x = ? + \int _ { b } ^ { \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$$

Accepted Answer

A)  $\int _ { - \infty } ^ { 2 } \frac { 9 } { x ^ { 2 } + 10 } d x$ 
B)  $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } - 9 } d x$ 
C)  $\int _ { - \infty } ^ { b } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
D)  $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
E)  none of these 
A)  $\int _ { - \infty } ^ { 2 } \frac { 9 } { x ^ { 2 } + 10 } d x$ 
B)  $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } - 9 } d x$ 
C)  $\int _ { - \infty } ^ { b } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
D)  $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$ 
E)  none of these

Question 7

Find the integral.
$$\int x ^ { 5 } \ln x d x$$

Accepted Answer

A)  $\frac { 1 } { 6 } x ^ { 6 } ( 6 \ln x - 1 ) + C$ 
B)  $\frac { 1 } { 36 } x ^ { 6 } ( \ln x - 1 ) + C$ 
C)  $\frac { 1 } { 6 } x ^ { 5 } + \frac { 1 } { x } + C$ 
D)  $\frac { 1 } { 36 } x ^ { 6 } ( 6 \ln x - 1 ) + C$ 
A)  $\frac { 1 } { 6 } x ^ { 6 } ( 6 \ln x - 1 ) + C$ 
B)  $\frac { 1 } { 36 } x ^ { 6 } ( \ln x - 1 ) + C$ 
C)  $\frac { 1 } { 6 } x ^ { 5 } + \frac { 1 } { x } + C$ 
D)  $\frac { 1 } { 36 } x ^ { 6 } ( 6 \ln x - 1 ) + C$

Question 8

Evaluate the integral. $$\int _ { x / 3 } ^ { x / 2 } 2 \cot ^ { 2 } x d x$$

Accepted Answer

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

Question 9

Use long division to evaluate the integral.
$$\int \frac { x ^ { 2 } } { x + 7 } d x$$

Accepted Answer

A)  $\frac { 1 } { 2 } ( x - 21 ) ( x + 7 ) + 49 \ln | x + 7 | + C$ 
B)  $\frac { 1 } { 2 } ( x + 21 ) ( x + 7 ) - 49 \ln | x + 7 | + C$ 
C)  $\frac { x ^ { 2 } } { 2 } - 14 x - 147 + \ln | x + 7 | + C$ 
D)  $\frac { x ^ { 2 } } { 2 } - 14 x + 147 + 49 \ln | x + 7 | + C$ 
E)  $\frac { x ^ { 2 } } { 2 } + 2 x + \ln | x + 49 | + C$ 
A)  $\frac { 1 } { 2 } ( x - 21 ) ( x + 7 ) + 49 \ln | x + 7 | + C$ 
B)  $\frac { 1 } { 2 } ( x + 21 ) ( x + 7 ) - 49 \ln | x + 7 | + C$ 
C)  $\frac { x ^ { 2 } } { 2 } - 14 x - 147 + \ln | x + 7 | + C$ 
D)  $\frac { x ^ { 2 } } { 2 } - 14 x + 147 + 49 \ln | x + 7 | + C$ 
E)  $\frac { x ^ { 2 } } { 2 } + 2 x + \ln | x + 49 | + C$

Question 10

Evaluate the integral using an appropriate trigonometric substitution. $$\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$$

Accepted Answer

The answer of Evaluate the integral using an appropriate trigonometric...

Question 11

Use a table of integrals to evaluate the integral. $$\int x \sqrt { 2 + 2 x } d x$$

Accepted Answer

A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
A)  $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
B)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$ 
C)  $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$ 
D)  $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$

Question 12

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....

Question 13

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

Accepted Answer

A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
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)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
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 14

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Accepted Answer

The answer of Make a substitution to express the integrand...

Question 15

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with $n = 12$ and the formula $R = \frac { 60 D } { \int _ { 0 } ^ { 24 } C ( t ) d t }$ to estimate the person's cardiac output, where $D$ is the quantity of dye injected in milligrams, $C ( t )$ is the concentration of the dye in the aorta, and $R$ is measured in liters per minute. Round to one decimal place.
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{t} & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 & 24 \
\hline \boldsymbol{C}(\boldsymbol{t}) & 0 & 0 & 2.6 & 6.3 & 9.7 & 7.5 & 4.5 & 3.5 & 2.2 & 0.6 & 0.3 & 0.1 & 0 \
\hline
\end{array}$

Accepted Answer

6.5 liters

Question 16

Evaluate the integral. $$\int \frac { \cos x } { 25 + \sin ^ { 2 } x } d x$$

Accepted Answer

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

Question 17

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

Accepted Answer

A)  $\frac { 1 } { 7 } \tan ^ { 7 } x + \frac { 2 } { 5 } \tan ^ { 5 } x + \frac { 1 } { 3 } \tan ^ { 3 } x + C$ 
B)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
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)  $\frac { 1 } { 5 } \tan ^ { 5 } x + \frac { 2 } { 3 } \tan ^ { 3 } x + \tan x + C$ 
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 18

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 19

Find the integral. $$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$$

Accepted Answer

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

Find the integral. $\int \frac { d x } { x ( x - 3 ) }$

Evaluate the integral or show that it is divergent. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

Evaluate the integral. $\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$

Evaluate the integral. $\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x$

Find the integral. $\int x ^ { 5 } \ln x d x$

Evaluate the integral. $\int _ { x / 3 } ^ { x / 2 } 2 \cot ^ { 2 } x d x$

Use long division to evaluate the integral. $\int \frac { x ^ { 2 } } { x + 7 } d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$

Use a table of integrals to evaluate the integral. $\int x \sqrt { 2 + 2 x } d x$

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

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

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Evaluate the integral. $\int \frac { \cos x } { 25 + \sin ^ { 2 } x } d x$

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

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.

Find the integral. $\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$

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

Exam 7: Techniques of Integration

Find the integral. ∫dxx(x−3)\int \frac { d x } { x ( x - 3 ) }∫x(x−3)dx​

Evaluate the integral or show that it is divergent. ∫−∞∞5dx4x2+4x+5\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }∫−∞∞​4x2+4x+55dx​

Evaluate the integral. ∫x/6x/33ln⁡(tan⁡x)7sin⁡xcos⁡xdx\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x∫x/6x/3​7sinxcosx3ln(tanx)​dx

Determine whether the improper integral converges or diverges, and if it converges, find its value. ∫π/25π/6cos⁡x1−sin⁡xdx\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x∫π/25π/6​1−sinx​cosx​dx

Evaluate the integral. ∫6e3x−exdx\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x∫e3x−ex6​dx

Find the integral. ∫x5ln⁡xdx\int x ^ { 5 } \ln x d x∫x5lnxdx

Evaluate the integral. ∫x/3x/22cot⁡2xdx\int _ { x / 3 } ^ { x / 2 } 2 \cot ^ { 2 } x d x∫x/3x/2​2cot2xdx

Use long division to evaluate the integral. ∫x2x+7dx\int \frac { x ^ { 2 } } { x + 7 } d x∫x+7x2​dx

Evaluate the integral using an appropriate trigonometric substitution. ∫13x2−1x4dx\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x∫13​x4x2−1​​dx

Use a table of integrals to evaluate the integral. ∫x2+2xdx\int x \sqrt { 2 + 2 x } d x∫x2+2x​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. ∫tan⁡2xsec⁡6xdx\int \tan ^ { 2 } x \sec ^ { 6 } x d x∫tan2xsec6xdx

Make a substitution to express the integrand as a rational function and then evaluate the integral. ∫425xx−100dx\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x∫425​x−100x​​dx Round the answer to four decimal places.

Evaluate the integral. ∫cos⁡x25+sin⁡2xdx\int \frac { \cos x } { 25 + \sin ^ { 2 } x } d x∫25+sin2xcosx​dx

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

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.

Find the integral. ∫x3−3x2+6x−2x3−2x2+xdx\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x∫x3−2x2+xx3−3x2+6x−2​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 integral. $\int \frac { d x } { x ( x - 3 ) }$

Evaluate the integral or show that it is divergent. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

Evaluate the integral. $\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { \pi / 2 } ^ { 5 \pi / 6 } \frac { \cos x } { \sqrt { 1 - \sin x } } d x$

Evaluate the integral. $\int \frac { 6 } { e ^ { 3 x } - e ^ { x } } d x$

Find the integral. $\int x ^ { 5 } \ln x d x$

Evaluate the integral. $\int _ { x / 3 } ^ { x / 2 } 2 \cot ^ { 2 } x d x$

Use long division to evaluate the integral. $\int \frac { x ^ { 2 } } { x + 7 } d x$

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 1 } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 1 } } { x ^ { 4 } } d x$

Use a table of integrals to evaluate the integral. $\int x \sqrt { 2 + 2 x } d x$

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

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

Make a substitution to express the integrand as a rational function and then evaluate the integral. $\int _ { 4 } ^ { 25 } \frac { \sqrt { x } } { x - 100 } d x$ Round the answer to four decimal places.

Evaluate the integral. $\int \frac { \cos x } { 25 + \sin ^ { 2 } x } d x$

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

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.

Find the integral. $\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 6 x - 2 } { x ^ { 3 } - 2 x ^ { 2 } + x } d x$