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evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) dxx2 10x

User Comecme
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I assume you mean


\displaystyle \int (dx)/(x^2 + 10x)

Expand the integrand into partial fractions.


\frac1{x^2 + 10x} = \frac1{x(x+10)} = \frac ax + \frac b{x+10} \\\\ ~~~~ \implies 1 = a(x+10) + bx = (a+b)x + 10a \\\\ ~~~~ \implies \begin{cases}a+b=0\\10a=1\end{cases} \\\\ ~~~~ \implies a=\frac1{10},b=-\frac1{10}

Then


\displaystyle \int (dx)/(x^2 + 10x) = \frac1{10} \left(\int \frac{dx}x - \int (dx)/(x+10)\right) \\\\ ~~~~~~~~ = \frac1{10} \left(\ln|x| - \ln|x+10|\right) + C \\\\ ~~~~~~~~ = \boxed{\frac1{10} \ln\left|\frac x{x+10}\right| + C}

User Didjit
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