# How do you integrate f(x)=lnx/(1+x^2) using the quotient rule?

Oct 30, 2016

There is no quotient rule for integrating.

#### Explanation:

There is a quotient rule for differentiating, perhaps that is what was meant.

$\frac{d}{\mathrm{dx}} \left(\frac{u}{v}\right) = \frac{u ' v - u v '}{v} ^ 2$

In this case $u = \ln x$, so $u ' = \frac{1}{x}$

and $v = 1 + {x}^{2}$

$\frac{d}{\mathrm{dx}} \left(\frac{\ln x}{1 + {x}^{2}}\right) = \frac{\frac{1}{x} \left(1 + {x}^{2}\right) - \left(\ln x\right) \left(2 x\right)}{1 + {x}^{2}} ^ 2$

$= \frac{1 + {x}^{2} - 2 {x}^{2} \ln x}{x {\left(1 + {x}^{2}\right)}^{2}}$