29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫ (x²) / (4x³ + 7) dx

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{x^{2}}{4x^{3} + 7} \, dx\).
2. Look for a substitution that simplifies the integral. Notice that the denominator is \$4x^{3} + 7\(, and its derivative involves \)x^{2}\(. Set \)u = 4x^{3} + 7$.
3. Compute the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = 12x^{2}\), which implies \(du = 12x^{2} \, dx\).
4. Rewrite the integral in terms of \(u\) and \(du\). Since \(du = 12x^{2} \, dx\), then \(x^{2} \, dx = \frac{du}{12}\). Substitute into the integral to get \(\int \frac{1}{u} \cdot \frac{du}{12} = \frac{1}{12} \int \frac{1}{u} \, du\).
5. Integrate \(\frac{1}{u}\) with respect to \(u\), which is \(\ln|u| + C\). Finally, substitute back \(u = 4x^{3} + 7\) to express the answer in terms of \(x\).

Key Concepts

Substitution Method

The substitution method simplifies integrals by changing variables to make the integral easier to solve. Typically, you identify a part of the integrand whose derivative also appears elsewhere in the integral. For example, setting u = 4x³ + 7 helps transform the integral into a simpler form involving u.

Recognizing Derivatives within the Integrand

This involves spotting expressions in the integrand that correspond to the derivative of another function present in the integral. In this problem, the derivative of 4x³ + 7 is 12x², which is closely related to the numerator x², guiding the substitution choice and simplifying the integral.

Handling Absolute Values in Logarithmic Integrals

When integrating functions that result in logarithms, the argument of the logarithm must be positive, so absolute values are used to ensure this. Absolute values are included only when the expression inside the log can be negative, ensuring the integral is correctly defined over its domain.