19. 5x - 5 = 3x -9
2x = -4
x = -2
19. Answer is A
Answer:
![(\frac{x^{2}-25}{2})ln(5+x)-\frac{x^{2}}{4}+\frac{5x}{2}+C](https://tex.z-dn.net/?f=%28%5Cfrac%7Bx%5E%7B2%7D-25%7D%7B2%7D%29ln%285%2Bx%29-%5Cfrac%7Bx%5E%7B2%7D%7D%7B4%7D%2B%5Cfrac%7B5x%7D%7B2%7D%2BC)
Step-by-step explanation:
Ok, so we start by setting the integral up. The integral we need to solve is:
![\int x ln(5+x)dx](https://tex.z-dn.net/?f=%5Cint%20x%20ln%285%2Bx%29dx)
so according to the instructions of the problem, we need to start by using some substitution. The substitution will be done as follows:
U=5+x
du=dx
x=U-5
so when substituting the integral will look like this:
![\int (U-5) ln(U)dU](https://tex.z-dn.net/?f=%5Cint%20%28U-5%29%20ln%28U%29dU)
now we can go ahead and integrate by parts, remember the integration by parts formula looks like this:
![\int (pq')=pq-\int qp'](https://tex.z-dn.net/?f=%5Cint%20%28pq%27%29%3Dpq-%5Cint%20qp%27)
so we must define p, q, p' and q':
p=ln U
![p'=\frac{1}{U}dU](https://tex.z-dn.net/?f=p%27%3D%5Cfrac%7B1%7D%7BU%7DdU)
![q=\frac{U^{2}}{2}-5U](https://tex.z-dn.net/?f=q%3D%5Cfrac%7BU%5E%7B2%7D%7D%7B2%7D-5U)
q'=U-5
and now we plug these into the formula:
![\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\int \frac{\frac{U^{2}}{2}-5U}{U}dU](https://tex.z-dn.net/?f=%5Cint%20%28U-5%29lnUdU%3D%28%5Cfrac%7BU%5E%7B2%7D%7D%7B2%7D-5U%29lnU-%5Cint%20%5Cfrac%7B%5Cfrac%7BU%5E%7B2%7D%7D%7B2%7D-5U%7D%7BU%7DdU)
Which simplifies to:
![\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\int (\frac{U}{2}-5)dU](https://tex.z-dn.net/?f=%5Cint%20%28U-5%29lnUdU%3D%28%5Cfrac%7BU%5E%7B2%7D%7D%7B2%7D-5U%29lnU-%5Cint%20%28%5Cfrac%7BU%7D%7B2%7D-5%29dU)
Which solves to:
![\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\frac{U^{2}}{4}+5U+C](https://tex.z-dn.net/?f=%5Cint%20%28U-5%29lnUdU%3D%28%5Cfrac%7BU%5E%7B2%7D%7D%7B2%7D-5U%29lnU-%5Cfrac%7BU%5E%7B2%7D%7D%7B4%7D%2B5U%2BC)
so we can substitute U back, so we get:
![\int xln(x+5)dU=(\frac{(x+5)^{2}}{2}-5(x+5))ln(x+5)-\frac{(x+5)^{2}}{4}+5(x+5)+C](https://tex.z-dn.net/?f=%5Cint%20xln%28x%2B5%29dU%3D%28%5Cfrac%7B%28x%2B5%29%5E%7B2%7D%7D%7B2%7D-5%28x%2B5%29%29ln%28x%2B5%29-%5Cfrac%7B%28x%2B5%29%5E%7B2%7D%7D%7B4%7D%2B5%28x%2B5%29%2BC)
and now we can simplify:
![\int xln(x+5)dU=(\frac{x^{2}}{2}+5x+\frac{25}{2}-25-5x)ln(5+x)-\frac{x^{2}+10x+25}{4}+25+5x+C](https://tex.z-dn.net/?f=%5Cint%20xln%28x%2B5%29dU%3D%28%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%7D%2B5x%2B%5Cfrac%7B25%7D%7B2%7D-25-5x%29ln%285%2Bx%29-%5Cfrac%7Bx%5E%7B2%7D%2B10x%2B25%7D%7B4%7D%2B25%2B5x%2BC)
![\int xln(x+5)dU=(\frac{x^{2}-25}{2})ln(5+x)-\frac{x^{2}}{4}-\frac{5x}{2}-\frac{25}{4}+25+5x+C](https://tex.z-dn.net/?f=%5Cint%20xln%28x%2B5%29dU%3D%28%5Cfrac%7Bx%5E%7B2%7D-25%7D%7B2%7D%29ln%285%2Bx%29-%5Cfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cfrac%7B5x%7D%7B2%7D-%5Cfrac%7B25%7D%7B4%7D%2B25%2B5x%2BC)
![\int xln(x+5)dU=(\frac{x^{2}-25}{2})ln(5+x)-\frac{x^{2}}{4}+\frac{5x}{2}+C](https://tex.z-dn.net/?f=%5Cint%20xln%28x%2B5%29dU%3D%28%5Cfrac%7Bx%5E%7B2%7D-25%7D%7B2%7D%29ln%285%2Bx%29-%5Cfrac%7Bx%5E%7B2%7D%7D%7B4%7D%2B%5Cfrac%7B5x%7D%7B2%7D%2BC)
notice how all the constants were combined into one big constant C.
Answer:
The answer is 9.
Step-by-step explanation:
6x - 2 < 5x + 7
6x - 5x < 7 + 2
x < 9