# What is the antiderivative of  (5x)/(x^2+1) ?

Jan 10, 2017

$F \left(x\right) = \frac{5}{2} \ln \left({x}^{2} + 1\right) + C$

#### Explanation:

The primitive of a function can be calculated as its indefinite integral:

$F \left(x\right) = \int \frac{5 x}{{x}^{2} + 1} \mathrm{dx}$

This integral can be calculated easily noting that:

$d \left({x}^{2} + 1\right) = 2 x \mathrm{dx}$ so:

$\int \frac{5 x}{{x}^{2} + 1} \mathrm{dx} = \frac{5}{2} \int \frac{2 x \mathrm{dx}}{{x}^{2} + 1} = \frac{5}{2} \int \frac{d \left({x}^{2} + 1\right)}{{x}^{2} + 1} = \frac{5}{2} \ln \left({x}^{2} + 1\right) + C$