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.

∫(x

^{2}+2)(x-1)^{7}dxIm 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

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## 1 Expert Answer

Michael W. answered • 02/18/15

Tutor

New to Wyzant
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 u

^{7}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. x

^{2}+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 x^{2}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

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 (x

^{2}+2) we get (u+1)^{2}+2, expand brackets to get, u^{2}+2u+1 then add the 2 to get - u^{2}+2u+3.So we now have: (u

^{2}+2u+3) x u^{7}= u^{9}+2u^{8}+3u^{7}Now integrate this to get - u

^{10}/10 + 2u^{9}/9 +3u^{8}/8 + cThen 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 + cThanks 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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Mitiku D.

^{2}+2 and get (u+1)^{2}+2 = u^{2}+2u+1 +2 = u^{2}+2u+3................... you are better of without substitution. It's much easier to just multiply out the factors (x^{2}+2) and (x-1)^{7}and integrate02/18/15