Integral: dx(2x^2)/[sqrt(x^3+1)]

[MarkSA]

Hello,

Finally finished the practice review worksheet (75 integrals), and I have one final question:

1) Indefinite integral of: dx(2x^2)/[sqrt(x^3+1)]
Here is what I did:
u = x^3 + 1, du = 3x^2dx, 1/3*du=x^2dx
I can change the fraction to get something du = 2x^2dx and directly substitute, but my question is this: Can I pull the '2' in the 2x^2 of the original integral outside of the integral before doing this problem? I thought so, but when I solved the integral my answer came out wrong.
I get the right answer though if I pull out 1/2 instead of 2 to get:
1/2 * Indefinite integral of: dx(x^2)/[sqrt(x^3+1)]
Is this the correct way to do this in general? If it was 3x^2 instead, could I then pull out 1/3 and solve the integral as normal and get the correct answer?

Thank you

 

[o_O]

How did you get du = 3x²dx to du = 2x²dx? I think that's your mistake there.

 

[Deleted member 4993]

Hello,

Finally finished the practice review worksheet (75 integrals), and I have one final question:

1) Indefinite integral of: dx(2x^2)/[sqrt(x^3+1)]
Here is what I did:
u = x^3 + 1, du = 3x^2dx, 1/3*du=x^2dx
I can change the fraction to get something du = 2x^2dx and directly substitute, but my question is this: Can I pull the '2' in the 2x^2 of the original integral outside of the integral before doing this problem? I thought so, but when I solved the integral my answer came out wrong.
I get the right answer though if I pull out 1/2 instead of 2 to get:
1/2 * Indefinite integral of: dx(x^2)/[sqrt(x^3+1)]
Is this the correct way to do this in general? If it was 3x^2 instead, could I then pull out 1/3 and solve the integral as normal and get the correct answer?

Thank you

I cannot quite see what you are talking about.

Can plese show your detailed work?

 

[MarkSA]

Sorry, here is what I did:

1) Indefinite integral of: dx(2x^2)/[sqrt(x^3+1)]
Change to: 2 * (integral: dx(x^2)/[sqrt(x^3+1)])
Here is what I did:
u = x^3 + 1
du = 3x^2dx
1/3*du=x^2dx
So, move 1/3 to the outside of the integral and substitute gets me:
2 * 1/3 * integral: u^(-1/2)du
= 2 * 1/3 * 2u^(1/2) + C
= 4/3u^(1/2) + C
= 4/3(x^3+1)^(1/2) + C

But this isn't correct according to the answer sheet. I'm wondering if I went wrong with moving the 2 to the outside of the integral. I thought that was allowed. Is it not? Any ideas?

 

[daon]

But this isn't correct according to the answer sheet.

Your answer is correct. Assuming the key is correct, it is possible your professor took another approach and obtained an equvilant solution (although it may not look the same right away), in which case yours should differ from the answer sheet by a constant.

 

[MarkSA]

Ahhh, thank you. You are right about the answer being done a different way. I probably should have just differentiated my answer and verified it myself first.