Calculus/Integrate

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Expert: Alon Mandes - 4/5/2009


Question
hi can u please show me the steps to answer a few integrate questions...
x + 2x^3/(x^4 + X^2) dx
sin^2 x cosx dx
sin /1 - sin^2 t dt

Answer
∫sint/(1-sin²t) dt = ∫sint/(cos²t) dt . We know that
(1/cost)'=[-1/cos²t][-sint]=sint/(cos²t). Thus,
∫sint/(1-sin²t) dt = 1/cost = sect .

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∫sin²x*cosx dx = ∫(1-cos²x)*cosx dx= ∫(cosx-cos³x)dx .
We know from trigonometrical identities that cos³Ø=cos3Ø+3cosØ. Thus,
∫sin²x*cosx dx = ∫(cosx-cos3x-3cosx)dx = ∫(-2cosx-cos3x) dx =
-2∫cosx dx -  ∫cos3x dx = -2sinx - (⅓)sin3x .

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First of all , lets deal with the integrand
f(x)=x+(2x³)/(x^4+x²)
f(x)=x+(2x)/(x²+1)
We know that (x²+1)'=2x & [Ln(x²+1)]'=2x/(x²+1) . Thus,
∫x+(2x³)/(x^4+x²) dx = (½)x²+Ln(x²+1) .



ALon.

Calculus

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Alon Mandes

Expertise

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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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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