Evaluate the integrals in Exercises 53–76.
75. ∫y dy/√(1-y^4)

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{y \, dy}{\sqrt{1 - y^{4}}}\).
2. Consider a substitution to simplify the expression under the square root. Notice that the denominator involves \(1 - y^{4}\), which can be rewritten as \(1 - (y^{2})^{2}\), suggesting a substitution involving \(y^{2}\).
3. Let \(u = y^{2}\). Then, compute \(du\) in terms of \(dy\): since \(u = y^{2}\), we have \(du = 2y \, dy\), or equivalently, \(y \, dy = \frac{du}{2}\).
4. Rewrite the integral in terms of \(u\): substitute \(y \, dy\) with \(\frac{du}{2}\) and \(\sqrt{1 - y^{4}}\) with \(\sqrt{1 - u^{2}}\), so the integral becomes \(\int \frac{\frac{du}{2}}{\sqrt{1 - u^{2}}} = \frac{1}{2} \int \frac{du}{\sqrt{1 - u^{2}}}\).
5. Recognize that \(\int \frac{du}{\sqrt{1 - u^{2}}}\) is a standard integral whose antiderivative is \(\arcsin(u) + C\). After integrating, substitute back \(u = y^{2}\) to express the answer in terms of \(y\).

Key Concepts

Integration Techniques

Integration techniques involve methods such as substitution, integration by parts, and recognizing standard integral forms. For the given integral, substitution is often used to simplify the integrand into a more manageable form, enabling direct integration.

Substitution Method

The substitution method replaces a complicated expression with a single variable to simplify the integral. In this problem, setting a substitution like u = y² or u = y⁴ can transform the integral into a standard form, making it easier to evaluate.

Integrals Involving Radical Expressions

Integrals containing radicals, such as √(1 - y⁴), often require recognizing patterns or using trigonometric or hyperbolic substitutions. Understanding how to handle these radicals is essential to rewrite the integral into a solvable form.