1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
storchak [24]
3 years ago
10

Evaluate the integral of the quantity x divided by the quantity x to the fourth plus sixteen, dx . (2 points) one eighth times t

he inverse tangent of the quantity x squared divided by 4, plus C one fourth times the natural log of x to the 4th power plus 16, plus C one fourth times the inverse tangent of the quantity x squared divided by 4, plus C the product of negative one fourth and 1 over the quantity squared of x squared plus 16, plus C
Mathematics
1 answer:
Anika [276]3 years ago
4 0

Answer:

\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(\frac{x^2}{4}) + c

Step-by-step explanation:

Given

\int\limits {\frac{x}{x^4 + 16}} \, dx

Required

Solve

Let

u = \frac{x^2}{4}

Differentiate

du = 2 * \frac{x^{2-1}}{4}\ dx

du = 2 * \frac{x}{4}\ dx

du = \frac{x}{2}\ dx

Make dx the subject

dx = \frac{2}{x}\ du

The given integral becomes:

\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{x}{x^4 + 16}} \, * \frac{2}{x}\ du

\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{1}{x^4 + 16}} \, * \frac{2}{1}\ du

\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{x^4 + 16}} \,\ du

Recall that: u = \frac{x^2}{4}

Make x^2 the subject

x^2= 4u

Square both sides

x^4= (4u)^2

x^4= 16u^2

Substitute 16u^2 for x^4 in \int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{x^4 + 16}} \,\ du

\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{16u^2 + 16}} \,\ du

Simplify

\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{16}* \frac{1}{8u^2 + 8}} \,\ du

\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{2}{16}\int\limits {\frac{1}{u^2 + 1}} \,\ du

\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}\int\limits {\frac{1}{u^2 + 1}} \,\ du

In standard integration

\int\limits {\frac{1}{u^2 + 1}} \,\ du = arctan(u)

So, the expression becomes:

\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}\int\limits {\frac{1}{u^2 + 1}} \,\ du

\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(u)

Recall that: u = \frac{x^2}{4}

\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(\frac{x^2}{4}) + c

You might be interested in
At 12pm, the temperature was 57o F. By 5pm, the thermometer read 36o F. Assuming a constant change in temperature, calculate and
BaLLatris [955]

Answer:

Unit rate of change = -4.2°F/hour

Step-by-step explanation:

Assuming a constant rate of change, a relationship between the change of temperature as a function of time (in hours) can be described as follows:

r= \frac{(36^oF - 57^o F)}{17 - 12}\\r= \frac{-4.2^o F}{hour}

There is a temperature drop from 12pm to 5pm, which explains the rate of change being negative. This rate means that at every hour that goes by, the temperature drops by 4.2 degrees.

6 0
3 years ago
Y= -1/3x+2
FrozenT [24]
Plugging x in, we have -x/3+2, which is -2/3+2=1.3 (with the 3 repeating).

To graph it, try desmos.com if you need help :)
8 0
3 years ago
The dimensions of a rectangle are shown. What is the area of the rectangle?
Black_prince [1.1K]

Answer:

8x + 10

(I'm not if it's correct)

7 0
3 years ago
Help!!!!!!!!!!!!!!!!
Dmitriy789 [7]
The answer is 2/(k+2)

3 0
3 years ago
Which value correctly complete the statement? 6,501.5 km = _______ m
shtirl [24]
<span>A) 6,501,500  is the answer to your question :)</span>
7 0
3 years ago
Other questions:
  • Some seventh-grade students at Michaelson Middle School were surveyed to find the amount of time they spent doing homework over
    10·1 answer
  • Which is greater the 3ed multiple of 10 ot 6th multiple of 5 ​
    8·2 answers
  • What is the volume of the prism the units are 7m 24m and 21m
    5·1 answer
  • Simple, plz help! Need Plato answer
    9·2 answers
  • How many hours and minutes from noon to a quarter to four in the afternoon?
    9·1 answer
  • The box shown has a volume of 8 cubic inches.
    10·1 answer
  • Evaluate |-7| + 3|4|
    5·2 answers
  • A frog jumps across a pond. Its height, as a function of time, is modeled by y
    11·1 answer
  • From 93 books to 144 books
    15·1 answer
  • 1, 6, 13, 22, 33, 46, 61 what comes next???????
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!