Question

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

Answer 1

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}`