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2x-4-5x-3-6x-2-6x-12-x-2-2x-2-3-2-dx-A-very-nice-solution-2x-4-5x-3-6x-2-6x-12-x-2-2x-2-3-2-dx-f-x-x-2-2x-2-C-f-x-ax-3-bx-2-cx-d-f




Question Number 137938 by Ñï= last updated on 09/Apr/21
                   ∫((2x^4 +5x^3 +6x^2 +6x+12)/((x^2 +2x+2)^(3/2) ))dx=?  A very nice solution::  ∫((2x^4 +5x^3 +6x^2 +6x+12)/((x^2 +2x+2)^(3/2) ))dx=((f(x))/( (√(x^2 +2x+2))))+C  f(x)=ax^3 +bx^2 +cx+d  f(x)′=3ax^2 +2bx+c  f(x)′(x^2 +2x+2)−(x+1)f(x)=2x^4 +5x^3 +6x^2 +6x+12  2a=2    ⇒a=1  6a+2b−a−b=5     ⇒b=0  2c+4b−c−d=6   2c−d=12    ⇒c=6    d=0  ⇒f(x)=x^3 +6x  ∫((2x^4 +5x^3 +6x^2 +6x+12)/((x^2 +2x+2)^(3/2) ))dx=((x^3 +6x)/( (√(x^2 +2x+2))))+C
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{2}{x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{12}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{\mathrm{3}/\mathrm{2}} }{dx}=? \\ $$$${A}\:{very}\:{nice}\:{solution}:: \\ $$$$\int\frac{\mathrm{2}{x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{12}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{\mathrm{3}/\mathrm{2}} }{dx}=\frac{{f}\left({x}\right)}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}}+{C} \\ $$$${f}\left({x}\right)={ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d} \\ $$$${f}\left({x}\right)'=\mathrm{3}{ax}^{\mathrm{2}} +\mathrm{2}{bx}+{c} \\ $$$${f}\left({x}\right)'\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)−\left({x}+\mathrm{1}\right){f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{12} \\ $$$$\mathrm{2}{a}=\mathrm{2}\:\:\:\:\Rightarrow{a}=\mathrm{1} \\ $$$$\mathrm{6}{a}+\mathrm{2}{b}−{a}−{b}=\mathrm{5}\:\:\:\:\:\Rightarrow{b}=\mathrm{0} \\ $$$$\mathrm{2}{c}+\mathrm{4}{b}−{c}−{d}=\mathrm{6}\:\:\:\mathrm{2}{c}−{d}=\mathrm{12}\:\:\:\:\Rightarrow{c}=\mathrm{6}\:\:\:\:{d}=\mathrm{0} \\ $$$$\Rightarrow{f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{6}{x} \\ $$$$\int\frac{\mathrm{2}{x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{12}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{\mathrm{3}/\mathrm{2}} }{dx}=\frac{{x}^{\mathrm{3}} +\mathrm{6}{x}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}}+{C} \\ $$
Commented by SLVR last updated on 11/Apr/21
can you explain first step RHS
$${can}\:{you}\:{explain}\:{first}\:{step}\:{RHS} \\ $$

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