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Calculus/Integration using substitution

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Expert: Alon Mandes - 1/31/2009


Question
Use the substitution u=x+2 to find [integral of (x^3)/((x+2)^2)dx].

I am unsure how to use the value of u given or how to begin, as the denominator is not the derivative nor multiple of the derivitive of another value in the expression...

Thanks forany guidance!

Answer
∫(x^3)/[(x+2)^2] dx. Let's set u=x+2 . We may see that du=dx.
Now, we will switch x by u-2 & switch dx by du :
∫ [(u-2)^3] / [u^2]  du.
Now we know that (a+b)^3=a^3+3a^2b+3ab^2+b^3. So,
∫ [u^3-6u^2+24u-8] / [u^2]  du
= ∫u du -6∫du + 24∫(du/u) - 8∫du/u^2 = u^2/2 - 6u + 24Ln(u) + 8/u
=(1/2)(x+2)^2-6(x+2)+24Ln(x+2)+8/(x+2).

ALon.  

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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