Chloe W.

asked • 02/18/15

Integration by u substitution help

I need to solve the indefinite integral, however it must be by the u - sub method.  
 
∫(x2+2)(x-1)7dx
 
Im guessing this is not the best way, however by tutor tells me it can be done the u sub way. I just can't seen to see how. Would be grateful of any help you good people can offer.
 
chloe

Mitiku D.

Chloe, if you have u=x-1, as indicated by the answer below, you can say that x=u+1, and you substitute that in x2+2 and get (u+1)2+2 = u2+2u+1 +2 = u2+2u+3................... you are better of without substitution. It's much easier to just multiply out the factors (x2+2) and (x-1)7 and integrate
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02/18/15

Michael W.

Sorry, um, you want to multiply out (x-1)7 ???  And then multiply that by x2 + 2?
 
Ooooookay, let's get started!
 
(x - 1)(x - 1) = x2 - 2x + 1
(x2 - 2x +1)(x - 1) = x3 - x2 - 2x2 - 2x + x - 1 = x3 - 3x2 + 3x - 1
 
I think Chloe would be done with the u-substitution version by now, and we've still got four more powers to go.  :)
 
-- Michael
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02/18/15

1 Expert Answer

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Michael W. answered • 02/18/15

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Chloe,
 
So, in your typical substitution problem, you're looking to substitute u for the composite function (the function within a function).  In this case, that'd be x-1, and eventually, that'll end up being u7 when you make the substitution.
 
dx wouldn't be an issue here, either, because if u = x-1, then du is just dx.
 
But you still have the other piece floating around.  x2+2.  Normally, when you do a u-substitution, the other piece ends up being the derivative of u, so it disappears.  But that didn't happen here, because du was just dx, not anything with an x2 in it.  What to do, what to do.  :) 
 
Well, if u = x-1, then what's x in terms of u?  Can you substitute that for x, simplify, and end up with an integral you can handle?
 
Does that help?
 
-- Michael
 
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Chloe W.

Thank you Michael, like you said normally the piece left over cancels out with the derivative these are the ones i a m used to doing. So with what you have said i have had a go, can you tell me if i have got the correct answer and if not where i have gone wrong?
 
u = x-1, therefore x = u+1. So if we sub this into (x2+2) we get (u+1)2+2, expand brackets to get, u2+2u+1 then add the 2 to get - u2+2u+3.
 
So we now have:  (u2+2u+3) x u7  = u9+2u8+3u7
Now integrate this to get - u10/10 + 2u9/9 +3u8/8 + c
 
Then the final answer will be the above with (x-1) subbed back to replace the u.
 
so: (x-1)10/10 + 2(x-1)9/9 + 3(x-1)8/8 + c
 
Thanks Chloe.
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02/18/15

Michael W.

Looks right to me!  And yes, it seems like you found the integral using the substitution faster than we could have multiplied (x-1) seven times.  :)
 
Nice work.
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02/18/15

Chloe W.

Thank you so much Michael!! Feel a lot more confident now.
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02/18/15

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