1. This came up on an AP Calc AB practice test; it's supposed to be easy to integrate, but I don't know how to:

x^2(sqrt(x^3+1))

Read: x squared times the square root of (x cubed plus one).

Can anyone help? Don't just give the answer, please: give a fairly detailed explanation.

Thanks,
spt  2. ### Related Discussions:

3. Without actually trying it, it seems like integration by parts may be a good place to start. Edit: Or possibly using the substitution u=x^3, du=3x^2, I dunno. :P

Edit again: Lance answered while I was making the above edit.  4. This problem is prime for a u substitution:

u=x^3
then,
du=3*x^2*dx

now substitute to get:

(1/3)*sqrt(u+1)*du, if it is clear at this point what the integral will yield you can stop, but you can also do another substitution to make the answer more clear

w=u+1
dw=du

so, now you have: (1/3)*sqrt(w)*dw

integrate with respect to w and you get: (2/9)*w^(3/2)+C
now substitute u back in: (2/9)*(u+1)^(3/2)+C
now substitute x back in: (2/9)*(x^3+1)^(3/2)+C  5. MagiMaster,
I'll see if I can do the integration by parts.

Lance,
I'm not familiar with the 'dw' thing...I guess it's not covered in Calc AB.

Thanks!  6. Well, substitution is much simpler than integration by parts. Gerenally speaking, integration by parts is a last resort unless you recognize certain patterns in the equations.  7. w is the same as u is the same as x. I could have chosen any variable, its just a substitution.In fact, if I had done the u substitution in this way:

u=x^3+1

I would not have had to do the w substituion. the only difference between the variables is the domain the integration lives on.

for example if you were integrating the original function with respect to x from 2 to 4, then you would integrate the u substituted function with respect to u from 8 to 64. (because u=x^3 so 2^3 and 4^3).

and then if you were integrating the w function with respect to w, you would integrate from 9 to 65. (because w=u+1 so 8+1 and 64+1).  8. Nvm. just realised how old this topic is!  Bookmarks
 Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Terms of Use Agreement