Problem 42 - Biocalculus Calculus for the Life Sciences

Question

Find the general indefinite integral. \(\int \sec t(\sec t+\tan t) d t\)

Step-by-Step Solution

Answer

Break the integral into two parts: \( \int \sec^2 t \, dt + \int \sec t \tan t \, dt \).

Step 1: Expand the Integral

To find the integral \( \int \sec t(\sec t + \tan t) \, dt \), begin by expanding the expression inside:\[ \int \sec t(\sec t + \tan t) \, dt = \int \sec^2 t \, dt + \int \sec t \tan t \, dt. \]

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

Break the integral into two parts: \( \int \sec^2 t \, dt + \int \sec t \tan t \, dt \).

Key Concepts

Integration Techniques

In solving indefinite integrals, integrating using different strategies is crucial for simplifying and solving calculus problems. One fundamental strategy is to expand or simplify the expression inside the integral first.
This helps break down the problem into manageable parts. For the integral \( \int \sec t (\sec t + \tan t) \, dt \), expanding results in two separate integrals: \( \int \sec^2 t \, dt \) and \( \int \sec t \tan t \, dt \).
This type of expansion is an elementary technique used frequently because it makes complicated expressions easier to work with.

• Expanding simplifies terms and often converts the integral into a more recognizable form.
• Solving the problem can become more intuitive when dealing approached in smaller chunks.
• Common integration techniques beyond expanding might involve substitution or integration by parts, which are useful for more complex forms.

Trigonometric Integrals

The realm of trigonometric integrals focuses on expressions involving trigonometric functions like sine, cosine, secant, tangent, and their variations.
These integrals often require knowledge of trigonometric identities and rules for integration. In the exercise, the trigonometric integral consists of functions \( \sec^2 t \) and \( \sec t \tan t \).
Knowing the antiderivatives of these specific functions is key to solving the integral efficiently.

• For \(\int \sec^2 t \, dt\), the antiderivative is \( \tan t + C\), where \(C\) represents the constant of integration.
• For \(\int \sec t \tan t \, dt\), the result is \( \sec t + C\), relying on derivatives of trigonometric functions.

Mastery of these standard results speeds up solving trigonometric integrals significantly. Always recall basic integrations to keep the process straightforward.

Calculus Problems

Calculus problems encompass a broad range of topics, including derivatives, integrals, limits, and infinite series. Indefinite integrals focus on finding a general function from its derivative, without specific boundary values.
In this case, the task is finding indefinite integrals for a function involving \( \sec t \) and \( \tan t \), which are often encountered in problems related to curves defined by trigonometry.
Solving such problems requires a blend of understanding core calculus concepts and applying them to trigonometric contexts.

• Begin with recognizing the structure of the expression.
• Apply appropriate integration strategies and identities to reduce complexity.
• Always remember to add the constant of integration, \(C\), as it represents an arbitrary constant needed for indefinite integrals.

Working through calculus problems strengthens problem-solving abilities and deepens the understanding of applied mathematics, a crucial aspect for further studies in sciences and engineering.