Question

integral ln(3x+5)

Answer 1

why?

Answer 2

the program wouldnt let me answer you back. so how do i integrate 3x (1/3x+5)

Answer 3

\[\int\frac{3x}{3x+5}dx\]do long division on the ration

Answer 4

ratio*

Answer 5

is long divison the only way?

Answer 6

no, but it is the most direct and mechanical

Answer 7

can you do it the long division way?

Answer 8

no

Answer 9

I will show you, one sec...

Answer 10

so\[\frac{3x}{3x+5}=1-\frac5{3x+5}\]

Answer 11

another quicker, but more clever way\[\frac{3x}{3x+5}=\frac{3x+5-5}{3x+5}=\frac{3x+5}{3x+5}-\frac5{3x+5}=1-\frac5{3x+5}\]

Answer 12

so nowe you should be able to integrate\[\int1-\frac5{3x+5}dx\]

Answer 13

now*

Answer 14

is the integral now 1/2 x (-5x/3x^2+5x)

Answer 15

?
I'm not seeing what you did there...

Answer 16

what is with the one half?

Answer 17

i dont know. im really not understanding any of this

Answer 18

you were doing well up to now\[\int1-\frac5{3x+5}dx=\int dx-\int\frac5{3x+5}dx\]

Answer 19

\[\int dx=x\]obviously, and\[\int\frac5{3x+5}dx=?\]

Answer 20

hint: u-sub

Answer 21

u=3x+5

Answer 22

yes, now carry it out

Answer 23

du=3dx, du/3=dx

Answer 24

nice...

Answer 25

5x/u

Answer 26

is it x-1/3(5x/3x+5)

Answer 27

yes, sorry for the late reply

Answer 28

wait no that's wrong

Answer 29

\[\int\frac5{3x+5}dx\]\[u=3x+5\implies du=3dx\implies\frac{du}3=dx\]\[\frac53\int\frac{du}u=\frac53\ln u=\frac53\ln(3x+5)\]

Answer 30

so\[\int1-\frac5{3x+5}dx=x-\frac53\ln(3x+5)+C\]which makes our fianl answer\[\int\ln(3x+5)dx=x\ln(3x+5)-x+\frac53\ln(3x+5)+C\]\[=3(x+5)\ln(3x+5)-x+C\]

Answer 31

typo above\[\int\ln(3x+5)dx=x\ln(3x+5)-x+\frac53\ln(3x+5)+C\]\[=\frac13(3x+5)\ln(3x+5)-x+C\]