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.
∫ (1 / (cos² x tan x)) dx from π/3 to π/4

Verified step by step guidance

1. Start by rewriting the integrand \(\frac{1}{\cos^{2} x \tan x}\) in terms of sine and cosine functions to simplify the expression. Recall that \(\tan x = \frac{\sin x}{\cos x}\), so substitute this into the integrand.
2. After substitution, simplify the integrand: \(\frac{1}{\cos^{2} x \cdot \frac{\sin x}{\cos x}} = \frac{1}{\cos^{2} x} \cdot \frac{\cos x}{\sin x} = \frac{\cos x}{\cos^{2} x \sin x} = \frac{1}{\cos x \sin x}\).
3. Recognize that the integrand is now \(\frac{1}{\sin x \cos x}\). Use the identity \(\sin 2x = 2 \sin x \cos x\) to rewrite the integrand as \(\frac{1}{\sin x \cos x} = \frac{2}{\sin 2x}\).
4. Rewrite the integral with the new integrand: \(\int_{\pi/3}^{\pi/4} \frac{2}{\sin 2x} \, dx\). Consider a substitution to simplify the integral further, such as \(u = 2x\), which implies \(du = 2 dx\) or \(dx = \frac{du}{2}\).
5. Change the limits of integration accordingly: when \(x = \pi/3\), \(u = 2 \cdot \pi/3 = \frac{2\pi}{3}\); when \(x = \pi/4\), \(u = 2 \cdot \pi/4 = \frac{\pi}{2}\). Substitute into the integral and simplify before proceeding to integrate \(\csc u\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify integrals by rewriting expressions in more manageable forms, such as converting tan x and cos² x into sine and cosine functions. Using identities like tan x = sin x / cos x helps transform the integral into a solvable form.

Substitution Method

The substitution method involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can be rewritten in terms of this variable, making it easier to evaluate. This technique is especially useful when the integral contains composite functions or products of functions and their derivatives.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two specific points, called limits of integration. Evaluating a definite integral requires finding the antiderivative and then applying the Fundamental Theorem of Calculus by substituting the upper and lower limits. Understanding how to handle these limits is essential for obtaining the final numerical value.