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
Choose the correct simplification of the expression 9 over h to the power of negative 3
Vitek1552 [10]
Remove the negative exponent by rewriting x^-3 as 1/x^3
(9)/(1/x^3)
multiply the numerator by the reciprocal of the denominator
9x^3
8 0
3 years ago
Read 2 more answers
Un hacendado a comprado doble numero de vacas que de bueyes, por cada vaca pago 70 dólares y por cada buey 85 dólares, si el imp
Oduvanchick [21]

Responder:

Número de bueyes = 12

Número de vacas = 24

Explicación paso a paso:

Dado que :

Número de vacas compradas = 2 veces el número de bueyes;

Sea número de bueyes = y

Número de vacas = 2 años

Costo de la vaca = 70

Costo de bueyes = 85

Costo total = 2700

Por lo tanto;

(2 años * 70) + (y * 85) = 2700

140 años + 85 años = 2700

225 años = 2700

225 años / 225 = 2700/225

y = 12

Número de bueyes = y = 12

Número de vacas = 2y = 2 * 12 = 24

5 0
2 years ago
Split 63 into the ratio 4:3
ladessa [460]

Answer:

36 and 27

Step-by-step explanation:

4+3=7

63/7=9

9x4=36

9x3=27

4 0
2 years ago
Read 2 more answers
Solve for x.<br> (17x-23% 551
Rasek [7]

Answer:

2=|3x|

Step-by-step explanation:

6 0
3 years ago
Solve.
Burka [1]

Answer:

48*F

Step-by-step explanation:

You know that the daytime high temp was 13 higher than the given low of 35.

Thus your equation would be as follows:

35 + 13 = x

35 + 13 = 48

x =  48

hope this helps you out.

good luck and god bless!

Jeremy

4 0
3 years ago
Other questions:
  • Does anyone know the answer??
    8·1 answer
  • Using 50 boxes of nails a carpenter was able to finish 450 bird houses. Write an equation that can be used to express the relati
    12·1 answer
  • Factor the expression 6x^2 + 5x + 1
    10·2 answers
  • To display The number of female members in band class over the past five years use a
    6·1 answer
  • Help ASAP PLEASE AND SHOW HOW YOU GOT THE ANSWER
    14·1 answer
  • Find the value of x in each case:
    9·2 answers
  • Find the value of given expression<br><br><img src="https://tex.z-dn.net/?f=%20%5Csqrt%7B1221%20%5Ctimes%201221%7D%20" id="TexFo
    6·2 answers
  • My last question please help I keep getting it wrong!!​
    7·2 answers
  • The store had a special on apples, 4 pounds for $2.80. The store's special on oranges was 3 pounds for $2.58. Which of the follo
    7·2 answers
  • The local gym holds two 45-minute workout sessions and three 30-minute sessions each week. Judy attended all of the sessions thi
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!