Substitute
and
. Then the integral transforms to
![\displaystyle \int \frac{x\,dx}{(x^2+4)^3} = \frac12 \int \frac{du}{u^3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bx%5C%2Cdx%7D%7B%28x%5E2%2B4%29%5E3%7D%20%3D%20%5Cfrac12%20%5Cint%20%5Cfrac%7Bdu%7D%7Bu%5E3%7D)
Apply the power rule.
![\displaystyle \int \frac{du}{u^3} = -\dfrac1{2u^2} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bdu%7D%7Bu%5E3%7D%20%3D%20-%5Cdfrac1%7B2u%5E2%7D%20%2B%20C)
Now put the result back in terms of
.
![\displaystyle \int \frac{x\,dx}{(x^2+4)^3} = \frac12 \left(-\dfrac1{2u^2} + C\right) = -\dfrac1{4u^2} + C = \boxed{-\dfrac1{4(x^2+4)^2} + C}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bx%5C%2Cdx%7D%7B%28x%5E2%2B4%29%5E3%7D%20%3D%20%5Cfrac12%20%5Cleft%28-%5Cdfrac1%7B2u%5E2%7D%20%2B%20C%5Cright%29%20%3D%20-%5Cdfrac1%7B4u%5E2%7D%20%2B%20C%20%3D%20%5Cboxed%7B-%5Cdfrac1%7B4%28x%5E2%2B4%29%5E2%7D%20%2B%20C%7D)
Suppose that some value, c, is a point of a local minimum point.
The theorem states that if a function f is differentiable at a point c of local extremum, then f'(c) = 0.
This implies that the function f is continuous over the given interval. So there must be some value h such that f(c + h) - f(c) >= 0, where h is some infinitesimally small quantity.
As h approaches 0 from the negative side, then:
![\frac{f(c + h) - f(c)}{h} \leq 0 \text{, where h is approaching 0 from the negative side}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28c%20%2B%20h%29%20-%20f%28c%29%7D%7Bh%7D%20%5Cleq%200%20%5Ctext%7B%2C%20where%20h%20is%20approaching%200%20from%20the%20negative%20side%7D)
As h approaches 0 from the positive side, then:
![\frac{f(c + h) - f(c)}{h} \geq 0](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28c%20%2B%20h%29%20-%20f%28c%29%7D%7Bh%7D%20%5Cgeq%200)
Thus, f'(c) = 0
Answer:
33. D.1/3
34. D. y= x+4
35. Slope= -2/3 Y-intercept= -6
Step-by-step explanation:
For number 33. D. 1/3 is the correct answer.
<em>Start at the point (0,2) Then, Go up 1 and Right 3. </em>
<em>As a result, you land in the point (3,3) This proves that the slope is 1/3.</em>
Note: the line is increasing and going upwards, so the slope is positive.
For number 34. D. y=x+4 is the correct answer.
<em>When writing an equation of a line, the correct format is y=mx+b. We are given both m and b. So all we have to do is plug these values into the equation y=mx+b. As a result, we get y=1x+4, which also means y=x+4</em>
For number 35. The Correct answer is: Slope= -2/3 Y-intercept= -6
<em>For this question I can't see all the answer choices but I know that the Slope is -2/3 and the Y-intercept is -6. Here's why: First you need to transform the equation 2x+3y=-18 into y=mx+b format. To do this, subtract -2x from both sides of the equation and you are left with 3y=-18-2x. You can flip -18 and -2x which makes the equation 3y=-2x-18. The next step is to divide both sides of the equation by 3, which gives you y=-2/3x-6. This means </em>Slope= -2/3 and Y-intercept= -6
Answer:
1.25x+2,3
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Step-by-step explanation: