How do I solve this problem step by step. -10( -4

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

Anika [276]

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

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

4 0

2 years ago

Ram and his sister Kalpana win a prize of Rs 2000. They decide to share the prize in the ratio of their ages. Ram is 15 years ol

V125BC [204]

9514 1404 393

Answer:

Ram -- ₹1200
Kalpana -- ₹800

Step-by-step explanation:

Their age total is 15+10=25, so Ram will get 15/25 = 0.6 of the total. Kalpana will get the remaining 0.4 of the total.

Ram receives 0.6 × ₹2000 = ₹1200

Kalpana receives 0.4 × ₹2000 = ₹800

8 0

2 years ago

STEP BY STEP PLEASE

Soloha48 [4]

We can find the side length of square 3 by dividing by 4, which is 9.

Then, we find the side length of square 2 by dividing it by 4, which is 12.

To find the AREA of square 1, we do a^2+b^2=c^2. This is basically adding up area of square 1 and square 2 to get square 3.

a^2+b^2=c^2
9^2+12^2=c^2
81+144=225

So the area is 225 units.

6 0

2 years ago

Read 2 more answers

A used car has a value of 15,250 when it’s purchased in 2012. The value of the car decreases at a rate of 7.5% per year

kotegsom [21]

p=15250

r=7.5/100=0.075

y = 15250*(1 - 0.075)^x

after 8 years it would be

x = 8

y = 15250*(1 - 0.075)^8

y = $8,173.42

after 8 years the value of the car is going to be $8,173.42

8 0

2 years ago

0.004 of all math majors at a certain university double major in music. Express this decimal as a percent.

RUDIKE [14]

I think the answer is 4%

3 0

2 years ago

How do I solve this problem step by step. -10( -4 - 2j