Question

can someone help me find the solution to this integration: ((5x+8)^1/2+11)/(sqrt5x+8)

Answer 1

one more time. is this
\[\int \frac{\sqrt{5x+8}+11}{\sqrt{5x+8}}dx=\int dx+\int\frac{11}{\sqrt{5x+8}}dx\]

Answer 2

if so, easy answer is
\[x+\frac{22}{5}\sqrt{5x+8}\]

Answer 3

yes that is the problem

Answer 4

then that is the answer. easy more or less

Answer 5

so, if i take the derivative of that answer i should get the orginal problem back right

Answer 6

comes with my personal guarantee

Answer 7

, how do i compsate for the chain rule here, and in general with working with integrals

Answer 8

i am not sure what "compensate for the chain rule" means. wanna take the derivative and see what we get?

Answer 9

okay

Answer 10

the derivative of x is 1. that was easy

Answer 11

okay

Answer 12

the derivative of
\[\sqrt{5x+8}\] is
\[\frac{5}{\sqrt{5x+8}}\]

Answer 13

no that is wrong

Answer 14

the derivative of
\[\sqrt{5x+8}\] is
\[\frac{5}{2\sqrt{x+8}}\]

Answer 15

how is that

Answer 16

so we have

\[1+\frac{22}{5}\times \frac{5}{2\sqrt{5x+8}}\]

Answer 17

the chain rule.

Answer 18

oh i got it

Answer 19

how is what? ok the derivative of
\[\sqrt{\text{something}}\] is \[\frac{ \text{derivative of something}}{2\sqrt{\text{something}}}\]

Answer 20

so we have
\[1+\frac{11}{\sqrt{5x+8}}\] and i'll be damned if i am doing the algebra, but if you put this over one denominator you will get your original integrand.

Answer 21

no, i got it

Answer 22

dont get mad

Answer 23

not mad at all. you just should have seen the algebra i did earlier

Answer 24

hours of it because today's problems all required a ton of algebra

Answer 25

i messed up, cause i didnt break up the term 22sqrt5x+8/5 into 22/5 *sqrt5x+8

Answer 26

so i just meant i will let you do this one.
hope the steps are clear

Answer 27

ahh.but it is ok yes?

Answer 28

might be alot to ask but, can youhelp me out mwith one optmization problem

Answer 29

sure as long as it is over before 4

Answer 30

judge judy time aka beer o'clock

Answer 31

shoot

Answer 32

The top and bottom margins of a poster are each 6cm. The side margins are each 4cm. If the printed area on the poster is fixed at 384 cm^2, find the dimensions of the poster with the smallest total area

Answer 33

total area includes margins right?

Answer 34

yes, i believe

Answer 35

ok inside margin call width x and height y, then we know
\[xy=384\] making
\[y=\frac{384}{x}\]

Answer 36

okay

Answer 37

whole area including margin will be
\[(x+8)(y+12)\]

Answer 38

now put
\[y=\frac{384}{x}\] to get total area
\[A(x)=(x+8)(\frac{384}{x}+12)\]

Answer 39

multiply this mess out, take derivative set = 0 and solve

Answer 40

i get
\[A(x)=384+12x+\frac{3072}{x}+96\]
\[A(x)=12x+\frac{3072}{x}+480\]
\[A'(x)=12-\frac{3072}{x^2}\]
\[A'(x)=\frac{12x^2-3072}{x^2}\]

Answer 41

solve
\[12x^2-3072=0\]
\[12x^2=3072\]
\[x^2=256\]
\[x=16\] and that is that!

Answer 42

thanks