If a number can be written or can be converted to p/q form, where p and q are integers and q is a non-zero number, then it is said to be rational and if it cannot be written in this form, then it is irrational.
Looks like the integral is
![\displaystyle\int\frac{x^4-2x^2+4x+1}{x^3-x^2-x+1}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7Bx%5E4-2x%5E2%2B4x%2B1%7D%7Bx%5E3-x%5E2-x%2B1%7D%5C%2C%5Cmathrm%20dx)
First simplify the integrand by long division (or however you like):
![\dfrac{x^4-2x^2+4x+1}{x^3-x^2-x+1}=x+1+\dfrac{4x}{x^3-x^2-x+1}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%5E4-2x%5E2%2B4x%2B1%7D%7Bx%5E3-x%5E2-x%2B1%7D%3Dx%2B1%2B%5Cdfrac%7B4x%7D%7Bx%5E3-x%5E2-x%2B1%7D)
Also notice that
![x^3-x^2-x+1=x^2(x-1)-(x-1)=(x^2-1)(x-1)=(x+1)(x-1)^2](https://tex.z-dn.net/?f=x%5E3-x%5E2-x%2B1%3Dx%5E2%28x-1%29-%28x-1%29%3D%28x%5E2-1%29%28x-1%29%3D%28x%2B1%29%28x-1%29%5E2)
Split the last term into partial fractions:
![\dfrac{4x}{(x+1)(x-1)^2}=\dfrac a{x+1}+\dfrac b{x-1}+\dfrac c{(x-1)^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B4x%7D%7B%28x%2B1%29%28x-1%29%5E2%7D%3D%5Cdfrac%20a%7Bx%2B1%7D%2B%5Cdfrac%20b%7Bx-1%7D%2B%5Cdfrac%20c%7B%28x-1%29%5E2%7D)
![4x=a(x-1)^2+b(x+1)(x-1)+c(x+1)](https://tex.z-dn.net/?f=4x%3Da%28x-1%29%5E2%2Bb%28x%2B1%29%28x-1%29%2Bc%28x%2B1%29)
![4x=(a+b)x^2+(c-2a)x+a-b+c](https://tex.z-dn.net/?f=4x%3D%28a%2Bb%29x%5E2%2B%28c-2a%29x%2Ba-b%2Bc)
![\implies\begin{cases}a+b=0\\c-2a=4\\a-b+c=0\end{cases}\implies a=-1,b=1,c=2](https://tex.z-dn.net/?f=%5Cimplies%5Cbegin%7Bcases%7Da%2Bb%3D0%5C%5Cc-2a%3D4%5C%5Ca-b%2Bc%3D0%5Cend%7Bcases%7D%5Cimplies%20a%3D-1%2Cb%3D1%2Cc%3D2)
So the integral is equivalent to
![\displaystyle\int\left(x+1-\frac1{x+1}+\frac1{x-1}+\frac2{(x-1)^2}\right)\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cleft%28x%2B1-%5Cfrac1%7Bx%2B1%7D%2B%5Cfrac1%7Bx-1%7D%2B%5Cfrac2%7B%28x-1%29%5E2%7D%5Cright%29%5C%2C%5Cmathrm%20dx)
![=\displaystyle\frac{x^2}2+x-\ln|x+1|-\ln|x-1|-\frac2{x-1}+C](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac%7Bx%5E2%7D2%2Bx-%5Cln%7Cx%2B1%7C-%5Cln%7Cx-1%7C-%5Cfrac2%7Bx-1%7D%2BC)
![=\displaystyle\frac{x^2}2+x-\ln|x^2-1|-\frac2{x-1}+C](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac%7Bx%5E2%7D2%2Bx-%5Cln%7Cx%5E2-1%7C-%5Cfrac2%7Bx-1%7D%2BC)
Answer:
1. 28:15
2. 3:13
Step-by-step explanation:
The ratios above have no common factor.
4:24 .
Common factor : 4
18:30
Common factor : 6
12:14
Common factor : 2
Answer:
6327
Step-by-step explanation:
2109 bpm. you want to know how many times in minutes
Well all you do is 2109*3=6327
Your answer is 6 gallons of gas.
Since the car can travel 32 miles per gallon(mpg), you would need to do 192 divided by 32 to get your answer. To verify it you can do 6(answer) times 32. Your answer would be 192, meaning you're right.