Question

\int \frac{12x^{3}-3x^{2}}{3x^{4}-x^{3}} dx

Explanation:

Step1: Let \( u = 3x^{4}-x^{3}\)

Differentiate \(u\) with respect to \(x\): \(du=(12x^{3}-3x^{2})dx\)

Step2: Substitute \(u\) and \(du\) into the integral

The integral \(\int\frac{12x^{3}-3x^{2}}{3x^{4}-x^{3}}dx\) becomes \(\int\frac{du}{u}\)

Step3: Integrate \(\frac{du}{u}\)

Using the formula \(\int\frac{1}{u}du=\ln|u| + C\) (where \(C\) is the constant of integration)

Step4: Substitute back \(u = 3x^{4}-x^{3}\)

We get \(\ln|3x^{4}-x^{3}|+C\)

Answer:

\(\ln|3x^{4}-x^{3}| + C\)