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How would you integrate 2x(x^2+1)^10 using integration by parts    

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chamenviraj | Student, Undergraduate | eNotes Newbie

Posted May 25, 2013 at 4:22 AM via web

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How would you integrate 2x(x^2+1)^10 using integration by parts

 

 

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pramodpandey | College Teacher | Valedictorian

Posted May 25, 2013 at 5:13 AM (Answer #1)

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We have given

`int2x(x^2+1)^10dx`

`substitute`

`x^2+1=t`

`2xdx=dt`

`int2x(x^2+1)dx=intt^10dt`

`=int1*t^10dt`

`=1*t^(10+1)/(10+1)-int((d)/(dt)(1))*t^11/11dt`

`=t^11/11-int0 dt`

`=t^11/11+c`

consider   f(t)=1 and g(t)= t^10

int(f(t)g(t)dt)=f(t)int(g(t)dt)-int{(f(t))'int(g(t)dt))dt}

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 25, 2013 at 10:46 AM (Answer #2)

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The method of integration by parts is not suitable in this case, since you cannot identify two different functions in the integrand 2x*(x^2 + 1)^10.

Since the functions 2x and (x^2 + 1)^10 are two polinomial functions and since derivative of x^2 + 1 yields 2x, you may use change of variable, instead of integration by parts.

The answer above ilustrates the substitution method and you should consider this method as the only one possible in this case.

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oldnick | Valedictorian

Posted May 26, 2013 at 9:17 PM (Answer #3)

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That is:  `1/11 (x^2+1) +C`

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