7/1 is an equivalent fraction for 70/10
Answer:
2005400sblackwell
2005400sblackwell
03/27/2020
Mathematics
Middle School
Javier worked the following hours in the month of October. What was the percent increase from week 2 to week 4?
Answer:
32%
Step-by-step explanation:
I don’t know. I’m on a test got the answer wrong and it showed me the right answer.
Write it as an equation:
X - 10 = 6x + 3
Now solve for x:
Subtract 3 from both sides:
x-13 = 6x
Subtract 1 x from both sides:
-13 = 5x
Divide both sides by 5:
x = -13/5
Answer:
You need more information. But typically multiplly by 100 so .1 would be 10 %
Step-by-step explanation:
Answer:
![\dfrac{1}{2}x^2\ln x - \dfrac{1}{4}x^2+\text{C}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20x%20-%20%5Cdfrac%7B1%7D%7B4%7Dx%5E2%2B%5Ctext%7BC%7D)
Step-by-step explanation:
<u>Fundamental Theorem of Calculus</u>
![\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Ctext%7Bf%7D%28x%29%5C%3A%5Ctext%7Bd%7Dx%3D%5Ctext%7BF%7D%28x%29%2B%5Ctext%7BC%7D%20%5Ciff%20%5Ctext%7Bf%7D%28x%29%3D%5Cdfrac%7B%5Ctext%7Bd%7D%7D%7B%5Ctext%7Bd%7Dx%7D%28%5Ctext%7BF%7D%28x%29%29)
If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a <u>constant of integration</u>.
![\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cbegin%7Bminipage%7D%7B4%20cm%7D%5Cunderline%7BIntegrating%20%24x%5En%24%7D%5C%5C%5C%5C%24%5Cdisplaystyle%20%5Cint%20x%5En%5C%3A%5Ctext%7Bd%7Dx%3D%5Cdfrac%7Bx%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%2B%5Ctext%7BC%7D%24%5Cend%7Bminipage%7D%7D)
Increase the power by 1, then divide by the new power.
Given <u>indefinite integral</u>:
![\displaystyle \int x \ln x \:\: \text{d}x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20x%20%5Cln%20x%20%5C%3A%5C%3A%20%5Ctext%7Bd%7Dx)
To integrate the given integral, use Integration by Parts:
![\boxed{\begin{minipage}{5 cm}\underline{Integration by parts} \\\\$\displaystyle \int u \dfrac{\text{d}v}{\text{d}x}\:\text{d}x=uv-\int v\: \dfrac{\text{d}u}{\text{d}x}\:\text{d}x$ \\ \end{minipage}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cbegin%7Bminipage%7D%7B5%20cm%7D%5Cunderline%7BIntegration%20by%20parts%7D%20%5C%5C%5C%5C%24%5Cdisplaystyle%20%5Cint%20u%20%5Cdfrac%7B%5Ctext%7Bd%7Dv%7D%7B%5Ctext%7Bd%7Dx%7D%5C%3A%5Ctext%7Bd%7Dx%3Duv-%5Cint%20v%5C%3A%20%5Cdfrac%7B%5Ctext%7Bd%7Du%7D%7B%5Ctext%7Bd%7Dx%7D%5C%3A%5Ctext%7Bd%7Dx%24%20%5C%5C%20%5Cend%7Bminipage%7D%7D)
![\text{Let }u=\ln x \implies \dfrac{\text{d}u}{\text{d}x}=\dfrac{1}{x}](https://tex.z-dn.net/?f=%5Ctext%7BLet%20%7Du%3D%5Cln%20x%20%5Cimplies%20%5Cdfrac%7B%5Ctext%7Bd%7Du%7D%7B%5Ctext%7Bd%7Dx%7D%3D%5Cdfrac%7B1%7D%7Bx%7D)
![\text{Let }\dfrac{\text{d}v}{\text{d}x}=x \implies v=\dfrac{1}{2}x^2](https://tex.z-dn.net/?f=%5Ctext%7BLet%20%7D%5Cdfrac%7B%5Ctext%7Bd%7Dv%7D%7B%5Ctext%7Bd%7Dx%7D%3Dx%20%5Cimplies%20v%3D%5Cdfrac%7B1%7D%7B2%7Dx%5E2)
Therefore:
![\begin{aligned}\displaystyle \int u \dfrac{dv}{dx}\:dx & =uv-\int v\: \dfrac{du}{dx}\:dx\\\\\implies \displaystyle \int x \ln x\:\:\text{d}x & = \ln x \cdot \dfrac{1}{2}x^2-\int \dfrac{1}{2}x^2 \cdot \dfrac{1}{x}\:\:dx\\\\& = \dfrac{1}{2}x^2\ln x -\int \dfrac{1}{2}x\:\:dx\\\\& = \dfrac{1}{2}x^2\ln x - \dfrac{1}{4}x^2+\text{C}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cdisplaystyle%20%5Cint%20u%20%5Cdfrac%7Bdv%7D%7Bdx%7D%5C%3Adx%20%26%20%3Duv-%5Cint%20v%5C%3A%20%5Cdfrac%7Bdu%7D%7Bdx%7D%5C%3Adx%5C%5C%5C%5C%5Cimplies%20%5Cdisplaystyle%20%5Cint%20x%20%5Cln%20x%5C%3A%5C%3A%5Ctext%7Bd%7Dx%20%26%20%3D%20%5Cln%20x%20%5Ccdot%20%5Cdfrac%7B1%7D%7B2%7Dx%5E2-%5Cint%20%5Cdfrac%7B1%7D%7B2%7Dx%5E2%20%5Ccdot%20%5Cdfrac%7B1%7D%7Bx%7D%5C%3A%5C%3Adx%5C%5C%5C%5C%26%20%3D%20%5Cdfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20x%20-%5Cint%20%5Cdfrac%7B1%7D%7B2%7Dx%5C%3A%5C%3Adx%5C%5C%5C%5C%26%20%3D%20%5Cdfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20x%20-%20%5Cdfrac%7B1%7D%7B4%7Dx%5E2%2B%5Ctext%7BC%7D%5Cend%7Baligned%7D)
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