Step-by-step Explanation:
We are going to solve this problem using integration by parts. We can write the integrand as
![\dfrac{4x}{x^2+4x+3} = \dfrac{4x}{(x+3)(x+1)}](https://tex.z-dn.net/?f=%5Cdfrac%7B4x%7D%7Bx%5E2%2B4x%2B3%7D%20%3D%20%5Cdfrac%7B4x%7D%7B%28x%2B3%29%28x%2B1%29%7D)
![\:\:\:\:\:\:\:\:\:\:\:\:\:= \dfrac{A}{x+3} + \dfrac{B}{x+1}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%20%5Cdfrac%7BA%7D%7Bx%2B3%7D%20%2B%20%5Cdfrac%7BB%7D%7Bx%2B1%7D)
![\:\:\:\:\:\:\:\:\:\:\:\:\:= \dfrac{Ax + A + Bx + 3B}{x^2+4x+3}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%20%5Cdfrac%7BAx%20%2B%20A%20%2B%20Bx%20%2B%203B%7D%7Bx%5E2%2B4x%2B3%7D)
![\:\:\:\:\:\:\:\:\:\:\:\:\:=\dfrac{(A+B)x + (A+3B)}{x^2+4x+3}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cdfrac%7B%28A%2BB%29x%20%2B%20%28A%2B3B%29%7D%7Bx%5E2%2B4x%2B3%7D)
Comparing the above term to the left hand side, we can see that
(A + B)x = 4x
A + 3B = 0
Solving for A and B, we find that A = 6 and B = -2 so our integrand becomes
![\dfrac{4x}{x^2+4x+3} = \dfrac{6}{x+3} - \dfrac{2}{x+1}](https://tex.z-dn.net/?f=%5Cdfrac%7B4x%7D%7Bx%5E2%2B4x%2B3%7D%20%3D%20%5Cdfrac%7B6%7D%7Bx%2B3%7D%20-%20%5Cdfrac%7B2%7D%7Bx%2B1%7D)
We can now easily integrate this expression as follows:
![\displaystyle \int{\dfrac{4x}{x^2+4x+3}}dx = 6\int{\dfrac{dx}{x+3}} - 2\int{\dfrac{dx}{x+1}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%7B%5Cdfrac%7B4x%7D%7Bx%5E2%2B4x%2B3%7D%7Ddx%20%3D%206%5Cint%7B%5Cdfrac%7Bdx%7D%7Bx%2B3%7D%7D%20-%202%5Cint%7B%5Cdfrac%7Bdx%7D%7Bx%2B1%7D%7D)
![\:\:\:\:\:\:\:\:\:\:\:\:\:= 6\ln|x+3| - 2\ln|x+1| + C](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%206%5Cln%7Cx%2B3%7C%20-%202%5Cln%7Cx%2B1%7C%20%2B%20C)
Answer:
(y^2)^5 · y^8 = y^10 * y^8 = y^18
Hope this helps and not to late. Have a great day!
2/5 Is the answer! Am pretty sure Hope this helped
7 books costs $14, etc. We have points in a line
(7,14)
(8,16)
(9,18)
They're all points on the line y=2x which just means it costs $2 a book. So 1 book costs $2, 2 cost $4, and 3 cost $6. We have three new points:
(1,2)
(2,4)
(3,6)
They're all on the same line, y=2x, just closer to the origin than the first three. We plot a point at x=1, y=2, etc.
Merry Christmas