What is $f(x) = int -2x^2+1/xdx$ if $f(2)=-1$ ?

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Answer 1 · 2017-02-17T16:06:34

$f(x)=(-2x^3)/3+ln|x|+13/3-ln2, or,$

$f(x)=1/3(13-2x^3)+ln|x/2|.$

$f(x)=int(-2x^2+1/x)dx=int-2x^2dx+int1/xdx$

$=-2intx^2dx+ln|x|=-2(x^(2+1)/(2+1))+ln|x|$

$:. f(x)=(-2x^3)/3+ln|x|+C........(star)$

But, $f(2)=-1 rArr ((-2)(2)^3)/3+ln|2|+C=-1$

$rArr C=16/3-1-ln2=13/3-ln2.$

Hence, by $(star)$ ,

$f(x)=(-2x^3)/3+ln|x|+13/3-ln2, or,$

$f(x)=1/3(13-2x^3)+ln|x/2|.$

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What is f(x) = int -2x^2+1/xdxf(x) = int -2x^2+1/xdx if f(2)=-1 f(2)=-1 ?