Question

any help is greatly appreciated!
Evaluate the following integral by making the given substitution:
int|x^2 sqrt(x^3+2) dx, u=x^3+2

Note: Any arbitrary constants used must be an upper-case "C".

Answer 1

Try to use the equation tab next time, that looks too messy

Answer 2

\[\int\limits x^2 \sqrt{x^3+2} dx, u=x^3+2\]

Answer 3

does that make it clearer? i know it has something to do with the substitution rule but cant work it out?

Answer 4

Oh ok. I see it now. Its a u-substitution problem, and since they already gave you what u should be, you need to find du which should be 3x^2

Answer 5

\[\frac{ 1 }{ 3 }du=x^2dx\]
so plug that in back into your integral

Answer 6

how do i plug it back in?

Answer 7

\[\int\limits \frac{ 1 }{ 3 } \sqrt{x^3+2} du\]

Answer 8

like that?

Answer 9

\[\frac{ \sqrt{x^3+2}}{ 3 }\]

Answer 10

Ok, ill explain to you quiickly.
Remember that we already substituted x^3+2 to be u, so your integral should = \[\frac{ 1 }{ 3 } \int\limits_{}^{}\sqrt{u}du\]

Answer 11

should that be the integral?

Answer 12

yes but how do i get the integral after that or is that the integral?

Answer 13

What i typed should be the integral. remember that u=x^3+2, and du is just x^2dx,
it is important for you to understand these steps, otherwise you'll struggle all through the remainder of cal 1

Answer 14

\[\frac{ 1 }{ 3 }\int\limits \sqrt(u) du\]

Answer 15

thats not the answer it said it was wrong?

Answer 16

yes i understand the subbing in steps i just dont understand why it keeps saying its wrong?

Answer 17

I never said that was the answer. that was just the first step, in getting to the answer lol

Answer 18

oh ok :) i thought it was the answer because i got the same earlier but also said it was wrong!

Answer 19

now if you understand how i got to the du differential, then all you need do is to just integrate sqrt(u)du

Answer 20

\[\frac{ 2u^3/2 }{ 3 }\]

Answer 21

\[\frac{ 2u^\frac{ 3 }{ 2 } }{3}\]

Answer 22

thats it integrated right?

Answer 23

Good. now plug in what ever we had as u back into your answer, and dont forget the 1/3 outside of the integral earlier

Answer 24

\[\frac{ 1 }{ 3 }\int\limits \frac{ 2(x^3+2)^\frac{ 3 }{ 2 } }{ 3 }\]

Answer 25

so that?

Answer 26

why still there is integral sign ?

Answer 27

You already did the integral lol, so take out the sign

Answer 28

Multiply your denominators, then dont forget to add +C

Answer 29

\[\frac{ 1 }{ 3 } \frac{ 2(x^3+2)^\frac{ 3 }{ 2 } }{ 3 } + C\]

Answer 30

so i just multiply that out now?

Answer 31

yes

Answer 32

\[\frac{ 8 }{ 9 } + C\]

Answer 33

thats what i got?

Answer 34

it says its wrong?

Answer 35

where did your x go?

Answer 36

i dont know? i put it into the calculator?

Answer 37

You are still gonna get an expression in terms of x as this is an indefinite integral

Answer 38

Your answer should be \[\frac{ 2(x^3+2)^\frac{ 3 }{ 2 } }{ 9 }+C\]

Answer 39

how did you work that out?

Answer 40

You already had the answer when you integrated, i just simplified by multiplying 1/3 across

Answer 41

oh ok :) thank you so much :)

Answer 42

You do need to practice though, calculus gets a lot more fun after this.

Answer 43

ya im starting to understand it abit better now :) thanks for the help :)

Answer 44

Sure no problem. Did it say the answer was right though?

Answer 45

ya it did

Answer 46

Ok. Have a good night

Answer 47

thanks :)