Answer:
the answer is 4/9
Step-by-step explanation:
1/3= 3/9
Answer:
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(\frac{x^2}{4}) + c](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%2Aarctan%28%5Cfrac%7Bx%5E2%7D%7B4%7D%29%20%2B%20c)
Step-by-step explanation:
Given
![\int\limits {\frac{x}{x^4 + 16}} \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx)
Required
Solve
Let
![u = \frac{x^2}{4}](https://tex.z-dn.net/?f=u%20%3D%20%5Cfrac%7Bx%5E2%7D%7B4%7D)
Differentiate
![du = 2 * \frac{x^{2-1}}{4}\ dx](https://tex.z-dn.net/?f=du%20%3D%202%20%2A%20%5Cfrac%7Bx%5E%7B2-1%7D%7D%7B4%7D%5C%20dx)
![du = 2 * \frac{x}{4}\ dx](https://tex.z-dn.net/?f=du%20%3D%202%20%2A%20%5Cfrac%7Bx%7D%7B4%7D%5C%20dx)
![du = \frac{x}{2}\ dx](https://tex.z-dn.net/?f=du%20%3D%20%5Cfrac%7Bx%7D%7B2%7D%5C%20dx)
Make dx the subject
![dx = \frac{2}{x}\ du](https://tex.z-dn.net/?f=dx%20%3D%20%5Cfrac%7B2%7D%7Bx%7D%5C%20du)
The given integral becomes:
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{x}{x^4 + 16}} \, * \frac{2}{x}\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20%2A%20%5Cfrac%7B2%7D%7Bx%7D%5C%20du)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{1}{x^4 + 16}} \, * \frac{2}{1}\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20%2A%20%5Cfrac%7B2%7D%7B1%7D%5C%20du)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{x^4 + 16}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%20%7B%5Cfrac%7B2%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%5C%20du)
Recall that: ![u = \frac{x^2}{4}](https://tex.z-dn.net/?f=u%20%3D%20%5Cfrac%7Bx%5E2%7D%7B4%7D)
Make
the subject
![x^2= 4u](https://tex.z-dn.net/?f=x%5E2%3D%204u)
Square both sides
![x^4= (4u)^2](https://tex.z-dn.net/?f=x%5E4%3D%20%284u%29%5E2)
![x^4= 16u^2](https://tex.z-dn.net/?f=x%5E4%3D%2016u%5E2)
Substitute
for
in ![\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{x^4 + 16}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%20%7B%5Cfrac%7B2%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%5C%20du)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{16u^2 + 16}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%20%7B%5Cfrac%7B2%7D%7B16u%5E2%20%2B%2016%7D%7D%20%5C%2C%5C%20du)
Simplify
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{16}* \frac{1}{8u^2 + 8}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%20%7B%5Cfrac%7B2%7D%7B16%7D%2A%20%5Cfrac%7B1%7D%7B8u%5E2%20%2B%208%7D%7D%20%5C%2C%5C%20du)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{2}{16}\int\limits {\frac{1}{u^2 + 1}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B2%7D%7B16%7D%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7Bu%5E2%20%2B%201%7D%7D%20%5C%2C%5C%20du)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}\int\limits {\frac{1}{u^2 + 1}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7Bu%5E2%20%2B%201%7D%7D%20%5C%2C%5C%20du)
In standard integration
![\int\limits {\frac{1}{u^2 + 1}} \,\ du = arctan(u)](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7Bu%5E2%20%2B%201%7D%7D%20%5C%2C%5C%20du%20%3D%20arctan%28u%29)
So, the expression becomes:
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}\int\limits {\frac{1}{u^2 + 1}} \,\ du](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7Bu%5E2%20%2B%201%7D%7D%20%5C%2C%5C%20du)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(u)](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%2Aarctan%28u%29)
Recall that: ![u = \frac{x^2}{4}](https://tex.z-dn.net/?f=u%20%3D%20%5Cfrac%7Bx%5E2%7D%7B4%7D)
![\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(\frac{x^2}{4}) + c](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7Bx%7D%7Bx%5E4%20%2B%2016%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%2Aarctan%28%5Cfrac%7Bx%5E2%7D%7B4%7D%29%20%2B%20c)
Answer:
3,
3x+5
Step-by-step explanation:
DIFFERENTIATE W.R.T. X
3
EVALUATE
3x+5
Slope=0.005/2.000=0.002
p - intercept = 974/1 = 974.00000
n- intercept = 974/-404 = 487/-202 = -2.41089
Oh and it is ok if you don’t have a lot of points I am great full for the points you are giving me and the answer is C)13,094 and can I have the brainliest since the other guy clearly did not answer the question right I only have one brainliest ☺️