76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
79. ∫ [sec t / (1 + sin t)] dt

Verified step by step guidance

1. Start by examining the integral \( \int \frac{\sec t}{1 + \sin t} \, dt \). Notice that the denominator contains \(1 + \sin t\), which suggests a trigonometric identity or substitution might simplify the expression.
2. Recall the Pythagorean identity and the conjugate expression: multiply numerator and denominator by the conjugate of the denominator, which is \(1 - \sin t\), to simplify the denominator using the difference of squares formula.
3. Rewrite the integral as \( \int \frac{\sec t (1 - \sin t)}{(1 + \sin t)(1 - \sin t)} \, dt = \int \frac{\sec t (1 - \sin t)}{1 - \sin^2 t} \, dt \).
4. Use the Pythagorean identity \(1 - \sin^2 t = \cos^2 t\) to simplify the denominator, so the integral becomes \( \int \frac{\sec t (1 - \sin t)}{\cos^2 t} \, dt \).
5. Express \(\sec t = \frac{1}{\cos t}\) and rewrite the integral as \( \int \frac{1 - \sin t}{\cos^3 t} \, dt \). From here, consider a substitution such as \(u = \sin t\) to proceed with integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They help simplify expressions and integrals by rewriting functions in more convenient forms, such as converting secant and sine into cosine and sine or using Pythagorean identities.

Substitution Method (Change of Variables)

The substitution method involves changing the variable of integration to simplify the integral. By choosing an appropriate substitution, the integral can be transformed into a more manageable form, often turning complicated expressions into rational functions suitable for further techniques like partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition breaks down a complex rational function into simpler fractions that are easier to integrate. This method is especially useful after substitution when the integral becomes a rational function, allowing straightforward integration of each simpler term.