You integrate it by parts:![\large\begin{array}{l} \mathsf{\displaystyle\int\!x\,e^{2x}\,dx\qquad\quad(i)}\\\\\\ \begin{array}{lcl} \mathsf{u=x}&~\Rightarrow~&\mathsf{du=dx}\\\\ \mathsf{dv=e^{2x}\,dx}&~\Leftarrow~&\mathsf{v=\dfrac{1}{2}\,e^{2x}} \end{array} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%5Cqquad%5Cquad%28i%29%7D%5C%5C%5C%5C%5C%5C%20%5Cbegin%7Barray%7D%7Blcl%7D%20%5Cmathsf%7Bu%3Dx%7D%26~%5CRightarrow~%26%5Cmathsf%7Bdu%3Ddx%7D%5C%5C%5C%5C%20%5Cmathsf%7Bdv%3De%5E%7B2x%7D%5C%2Cdx%7D%26~%5CLeftarrow~%26%5Cmathsf%7Bv%3D%5Cdfrac%7B1%7D%7B2%7D%5C%2Ce%5E%7B2x%7D%7D%20%5Cend%7Barray%7D%20%5Cend%7Barray%7D)
![\large\begin{array}{l} \mathsf{\displaystyle\int\!u\,dv=u\cdot v-\int\!v\,du}\\\\ \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=x\cdot \frac{1}{2}\,e^{2x}-\int\!\frac{1}{2}\,e^{2x}\,dx}\\\\ \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{2}\int\!e^{2x}\,dx}\\\\ \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{2}\int\!\dfrac{1}{2}\cdot 2e^{2x}\,dx}\\\\ \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{2}\cdot \frac{1}{2}\int\!e^{2x}\cdot 2\,dx} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21u%5C%2Cdv%3Du%5Ccdot%20v-%5Cint%5C%21v%5C%2Cdu%7D%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3Dx%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5C%2Ce%5E%7B2x%7D-%5Cint%5C%21%5Cfrac%7B1%7D%7B2%7D%5C%2Ce%5E%7B2x%7D%5C%2Cdx%7D%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21e%5E%7B2x%7D%5C%2Cdx%7D%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21%5Cdfrac%7B1%7D%7B2%7D%5Ccdot%202e%5E%7B2x%7D%5C%2Cdx%7D%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21e%5E%7B2x%7D%5Ccdot%202%5C%2Cdx%7D%20%5Cend%7Barray%7D)
![\large\begin{array}{l} \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{4}\int\!e^{2x}\cdot 2\,dx}\\\\ \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{4}\int\!e^w\,dw\qquad(w=2x)}\\\\ \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{4}\,e^w+C}\\\\\\ \therefore~~\boxed{\begin{array}{c} \mathsf{\displaystyle\int\!x\,e^{2x}\,dx=\frac{1}{2}\,x\,e^{2x}-\frac{1}{4}\,e^{2x}+C} \end{array}}\qquad\checkmark \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B4%7D%5Cint%5C%21e%5E%7B2x%7D%5Ccdot%202%5C%2Cdx%7D%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B4%7D%5Cint%5C%21e%5Ew%5C%2Cdw%5Cqquad%28w%3D2x%29%7D%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B4%7D%5C%2Ce%5Ew%2BC%7D%5C%5C%5C%5C%5C%5C%20%5Ctherefore~~%5Cboxed%7B%5Cbegin%7Barray%7D%7Bc%7D%20%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21x%5C%2Ce%5E%7B2x%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Cx%5C%2Ce%5E%7B2x%7D-%5Cfrac%7B1%7D%7B4%7D%5C%2Ce%5E%7B2x%7D%2BC%7D%20%5Cend%7Barray%7D%7D%5Cqquad%5Ccheckmark%20%5Cend%7Barray%7D)
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![\large\textsf{I hope it helps. :-)}](https://tex.z-dn.net/?f=%5Clarge%5Ctextsf%7BI%20hope%20it%20helps.%20%3A-%29%7D)
Tags: <em>integrate indefinite integral product by parts polynomial exponential composite substitution integral calculus</em>
<span>The formula for the volume of a cylinder is V=πr²h. The volume of a cylinder is three times the volume of a cone with the same radius and height.
Now, volume of cone is 1/3</span>πh
![r^{2}](https://tex.z-dn.net/?f=%20r%5E%7B2%7D%20)
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If the volume of a cone with the same height as a cylinder equals the volume of a cylinder, the equation for the radius of cone in terms of the radius of cylinder R, is equal to 3R.
We get this by equating the volume of cone and cylinder:
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![\pi R^{2} h = \frac{1}{3} \pi r^{2}h \\ \\ thus, r = 3R](https://tex.z-dn.net/?f=%20%20%5Cpi%20R%5E%7B2%7D%20h%20%3D%20%20%5Cfrac%7B1%7D%7B3%7D%20%20%5Cpi%20%20r%5E%7B2%7Dh%20%5C%5C%20%20%5C%5C%20%20thus%2C%20r%20%3D%203R)
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Answer:
Answer is 73.
Step-by-step explanation:
![73(\frac{4}{5} )^x](https://tex.z-dn.net/?f=73%28%5Cfrac%7B4%7D%7B5%7D%20%29%5Ex)
To find y intercept set x = 0
= ![73*1](https://tex.z-dn.net/?f=73%2A1)
Y-intercept is 73
Answer:
4 cm
Step-by-step explanation:
If 6 inches = 1 cm, and 24 inches divided 6 inches = 4, then it would be 4 cm.
Answer: Simple random sampling
Step-by-step explanation:
Given: A market researcher obtains a list of all streets in a town. She randomly samples 10 street names from the list, and then administers survey questions to every family living on those 10 streets.
Since she randomly samples street names , therefore the type of sampling is simple random sampling.
- A simple random sample is a sample that is a subset of the population the researcher surveyed selected in a way such that all the individuals in the population has an equal chance to be selected.