Question

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

Explanation:

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

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

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

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

Step3: Integrate \( \frac{1}{u} \)

We know that \( \int\frac{du}{u}=\ln|u| + C \) (where \( C \) is the constant of integration)

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

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

Answer:

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