Answer:
A) ![\int\frac{2x^2}{3}dx=\frac{2x^3}{9}+C](https://tex.z-dn.net/?f=%5Cint%5Cfrac%7B2x%5E2%7D%7B3%7Ddx%3D%5Cfrac%7B2x%5E3%7D%7B9%7D%2BC)
B) ![\int(5-x)^{23}dx=-\frac{(5-x)^{24}}{24}+C](https://tex.z-dn.net/?f=%5Cint%285-x%29%5E%7B23%7Ddx%3D-%5Cfrac%7B%285-x%29%5E%7B24%7D%7D%7B24%7D%2BC)
Step-by-step explanation:
A)
So we have the integral:
![\int\frac{2x^2}{3}dx](https://tex.z-dn.net/?f=%5Cint%5Cfrac%7B2x%5E2%7D%7B3%7Ddx)
First, remove the constant multiple:
![=\frac{2}{3}\int x^2\dx](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%7D%7B3%7D%5Cint%20x%5E2%5Cdx)
Use the power rule, where:
![\int x^ndx=\frac{x^{n+1}}{x+1}](https://tex.z-dn.net/?f=%5Cint%20x%5Endx%3D%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7Bx%2B1%7D)
Therefore:
![\frac{2}{3}\int x^2\dx\\=\frac{2}{3}(\frac{x^{2+1}}{2+1})](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B3%7D%5Cint%20x%5E2%5Cdx%5C%5C%3D%5Cfrac%7B2%7D%7B3%7D%28%5Cfrac%7Bx%5E%7B2%2B1%7D%7D%7B2%2B1%7D%29)
Simplify:
![=\frac{2}{3}(\frac{x^{3}}{3})](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%7D%7B3%7D%28%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%29)
And multiply:
![=\frac{2x^3}{9}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2x%5E3%7D%7B9%7D)
And, finally, plus C:
![=\frac{2x^3}{9}+C](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2x%5E3%7D%7B9%7D%2BC)
B)
We have the integral:
![\int(5-x)^{23}dx](https://tex.z-dn.net/?f=%5Cint%285-x%29%5E%7B23%7Ddx)
To solve, we can use u-substitute.
Let u equal 5-x. Then:
![u=5-x\\du=-1dx](https://tex.z-dn.net/?f=u%3D5-x%5C%5Cdu%3D-1dx)
So:
![\int(5-x)^{23}dx\\=\int-u^{23}du](https://tex.z-dn.net/?f=%5Cint%285-x%29%5E%7B23%7Ddx%5C%5C%3D%5Cint-u%5E%7B23%7Ddu)
Move the negative outside:
![=-\int u^{23}du](https://tex.z-dn.net/?f=%3D-%5Cint%20u%5E%7B23%7Ddu)
Power rule:
![=-(\frac{u^{23+1}}{23+1})](https://tex.z-dn.net/?f=%3D-%28%5Cfrac%7Bu%5E%7B23%2B1%7D%7D%7B23%2B1%7D%29)
Add:
![=-(\frac{u^{24}}{24})](https://tex.z-dn.net/?f=%3D-%28%5Cfrac%7Bu%5E%7B24%7D%7D%7B24%7D%29)
Substitute back 5-x:
![=-(\frac{(5-x)^{24}}{24})](https://tex.z-dn.net/?f=%3D-%28%5Cfrac%7B%285-x%29%5E%7B24%7D%7D%7B24%7D%29)
Constant of integration:
![=-\frac{(5-x)^{24}}{24}+C](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B%285-x%29%5E%7B24%7D%7D%7B24%7D%2BC)
And we're done!
Answer:
42
Step-by-step explanation:
the answer to life and the universe
Pythagorean theorem a^2+b^2=c^2
Answer:
133
Step-by-step explanation: