Use partial fractions to solve the integral.

{eq}\displaystyle \int \frac{ (5x^2-3) }{(x^3-x)} \ dx {/eq}

Question:

Simplifying the Integral Value with Logarithmic Terms:

The value of the integral consists of the logarithmic terms represented as {eq}\int {f\left( x \right)dx} = a\ln \left( {r\left( x \right)} \right) + \ln \left( {s\left( x \right)} \right) - \ln \left( {p\left( x \right)} \right) + C {/eq}. It is simplified using the logarithmic identities like {eq}\ln {m^n} = n\ln m {/eq}, {eq}\ln m + \ln n = \ln \left( {mn} \right) {/eq}, {eq}\ln m - \ln n = \ln \left( {\dfrac{m}{n}} \right) {/eq}, and so on. Using these identities, we get the simplified expression of the given integral value as {eq}\int {f\left( x \right)dx} = \ln \left( {\dfrac{{{{\left( {r\left( x \right)} \right)}^a} \cdot s\left( x \right)}}{{p\left( x \right)}}} \right) + C {/eq}.

Answer and Explanation: 1