Evaluate the integrals in Exercises 47–68.

∫₀^π/4 sec²x / (1 + 7 tan x)²/³ dx

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Identify the integral to solve: \(\int_0^{\pi/4} \frac{\sec^2 x}{(1 + 7 \tan x)^{2/3}} \, dx\).

Recognize that the presence of \(\sec^2 x\) and \(\tan x\) suggests using the substitution \(u = 1 + 7 \tan x\) to simplify the integral.

Compute the differential \(du\): since \(u = 1 + 7 \tan x\), then \(du = 7 \sec^2 x \, dx\), which implies \(\sec^2 x \, dx = \frac{du}{7}\).

Change the limits of integration from \(x\) to \(u\): when \(x = 0\), \(u = 1 + 7 \tan 0 = 1\); when \(x = \frac{\pi}{4}\), \(u = 1 + 7 \tan \frac{\pi}{4} = 1 + 7 = 8\).

Rewrite the integral in terms of \(u\): \(\int_1^8 \frac{1}{u^{2/3}} \cdot \frac{1}{7} \, du = \frac{1}{7} \int_1^8 u^{-2/3} \, du\), which is now a standard power integral.

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