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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Trigonometric Functions

This involves techniques to integrate functions containing trigonometric expressions like sec²x and tan x. Recognizing standard derivatives, such as the derivative of tan x being sec²x, helps simplify integrals and apply substitution methods effectively.

Substitution Method in Integration

Substitution is used to simplify integrals by changing variables, often turning complex expressions into more manageable forms. For this integral, letting u = tan x transforms the integral into a rational function of u, making it easier to evaluate.

Handling Fractional Exponents in Integrals

Integrals involving expressions raised to fractional powers require careful manipulation, often involving rewriting the integrand to a power form that can be integrated using power rules or substitution. Understanding how to work with fractional exponents is key to solving such integrals.

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