Question

Find the indefinite integral. $$ \int \frac{x^{2}}{3-x^{3}} d x $$

Short answer

The indefinite integral of the function \( \frac{x^{2}}{3-x^{3}} \) with respect to \( x \) is \( - \frac{1}{3} \ln |3 - x^{3}| + c \)

Step-by-step solution

Substitute the function

Set a new variable \( u = 3 - x^3 \). Then calculate the differential of \( u \), which is \( du = -3x^{2} dx \). To isolate \( dx \), express it as \( dx = - \frac{du}{3x^2} \).

Replace in integral

Replace \( x^{2} dx \) with \( - \frac{du}{3} \) in the integral and the function \( 3 - x^{3} \) with \( u \). This turns the integral into \( - \frac{1}{3} \int \frac{1}{u} du \).

Simplify the integral

Now the integral is simplified to one of the elementary form, that can be directly integrated which results in \( - \frac{1}{3} \ln |u| + c \), where \( c \) denotes the constant of integration.

Back substitute \( u \) with \( x \)

Now, replace \( u \) back by \( 3 - x^{3} \) from the Step 1. This results in \( - \frac{1}{3} \ln |3 - x^{3}| + c \).