The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{\sqrt{x}}{1 + x^{3}} \, dx\). Notice the hint suggests using the substitution \(u = x^{3/2}\), which is equivalent to \(u = x^{\frac{3}{2}}\).
2. Differentiate \(u\) with respect to \(x\) to find \(du\): Since \(u = x^{3/2}\), then \(\frac{du}{dx} = \frac{3}{2} x^{1/2}\). This implies \(du = \frac{3}{2} x^{1/2} dx\).
3. Rewrite the integral in terms of \(u\) and \(du\). From the expression for \(du\), solve for $x^{1/2} dx$: \(x^{1/2} dx = \frac{2}{3} du\). Also, express the denominator \(1 + x^{3}\) in terms of \(u\): since \(u = x^{3/2}\), then \(u^{2} = x^{3}\), so \(1 + x^{3} = 1 + u^{2}\).
4. Substitute these expressions back into the integral: replace \(\sqrt{x} dx\) with \(\frac{2}{3} du\) and \(1 + x^{3}\) with \(1 + u^{2}\). The integral becomes \(\int \frac{\sqrt{x}}{1 + x^{3}} dx = \int \frac{\frac{2}{3} du}{1 + u^{2}}\).
5. Now, the integral simplifies to \(\frac{2}{3} \int \frac{1}{1 + u^{2}} du\), which is a standard integral. Recognize that \(\int \frac{1}{1 + u^{2}} du = \arctan(u) + C\). After integrating, substitute back \(u = x^{3/2}\) to express the answer in terms of \(x\).

Key Concepts

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. By letting u equal a function of x, we rewrite the integral in terms of u and du, making it easier to integrate. This technique is especially useful when the integral contains composite functions.

Algebraic Manipulation of Expressions

Algebraic manipulation involves rewriting expressions to reveal simpler forms or to match substitution patterns. In this problem, expressing √x and x³ in terms of powers of x helps identify a suitable substitution. Recognizing and manipulating exponents is key to simplifying the integral before applying substitution.

Integration of Rational Functions

Integrals involving rational functions, where one polynomial is divided by another, often require substitution or partial fraction decomposition. Understanding how to handle integrals with expressions like 1/(1 + u) after substitution is essential. This knowledge helps in evaluating the integral once the substitution is applied.