The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (4r dr) / √(1 − r⁴)

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Recognize that the integral is of the form \(\int_0^1 \frac{4r}{\sqrt{1 - r^4}} \, dr\). The integrand involves a composite function with \(r^4\) inside the square root, suggesting a substitution related to \(r^2\) or \(r^4\).

Make the substitution \(u = r^2\). Then, compute the differential: \(du = 2r \, dr\), which implies \(r \, dr = \frac{du}{2}\). This will help simplify the integral.

Rewrite the integral in terms of \(u\): replace \(4r \, dr\) with \(4 \times \frac{du}{2} = 2 \, du\), and replace \(\sqrt{1 - r^4}\) with \(\sqrt{1 - u^2}\) since \(r^4 = (r^2)^2 = u^2\). The integral becomes \(\int_0^1 \frac{2}{\sqrt{1 - u^2}} \, du\).

Recognize that \(\int \frac{1}{\sqrt{1 - u^2}} \, du\) is the standard integral for \(\arcsin u\). Therefore, the integral simplifies to \(2 \int_0^1 \frac{1}{\sqrt{1 - u^2}} \, du = 2 [\arcsin u]_0^1\).

Evaluate the definite integral by substituting the limits into \(\arcsin u\): compute \(2 (\arcsin 1 - \arcsin 0)\). This will give the value of the original integral.

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