Questions: ∫ (3x^2)/(3x^3-2) dx

Solution Steps

To solve the integral \(\int \frac{3x^2}{3x^3 - 2} \, dx\), we can use the method of substitution. Notice that the derivative of the denominator \(3x^3 - 2\) is \(9x^2\), which is similar to the numerator \(3x^2\). This suggests a substitution where \(u = 3x^3 - 2\), and \(du = 9x^2 \, dx\). We can then adjust the integral accordingly and solve it in terms of \(u\).

Queremos calcular la integral \[ \int \frac{3x^2}{3x^3 - 2} \, dx. \] Para resolverla, utilizamos la sustitución \( u = 3x^3 - 2 \). Entonces, la derivada de \( u \) es \( du = 9x^2 \, dx \), lo que implica que \( dx = \frac{du}{9x^2} \).

Sustituyendo en la integral, obtenemos: \[ \int \frac{3x^2}{u} \cdot \frac{du}{9x^2} = \int \frac{1}{3} \cdot \frac{1}{u} \, du. \]

La integral de \( \frac{1}{u} \) es \( \log|u| \). Por lo tanto, tenemos: \[ \int \frac{1}{3} \cdot \frac{1}{u} \, du = \frac{1}{3} \log|u| + C. \] Sustituyendo \( u = 3x^3 - 2 \) de nuevo, obtenemos: \[ \frac{1}{3} \log|3x^3 - 2| + C. \]

Final Answer