Problem 41 - Biocalculus Calculus for the Life Sciences

Question

Find the general indefinite integral. \(\int\left(1+\tan ^{2} \alpha\right) d \alpha\)

Step-by-Step Solution

Answer

\(\tan \alpha + C\) is the general indefinite integral.

Step 1: Identify the Integration Formula

Recognize that the expression \(1 + \tan^2 \alpha\) is equivalent to \(\sec^2 \alpha\). This identity comes from the Pythagorean identity for tangent and secant functions: \(1 + \tan^2 \alpha = \sec^2 \alpha\).

Step 2: Replace the Original Function

Rewrite the integral using the identity from Step 1: \[\int (1 + \tan^2 \alpha) \, d\alpha = \int \sec^2 \alpha \, d\alpha\].

Step 3: Integrate the New Expression

Recall that the integral of \(\sec^2 \alpha\) with respect to \(\alpha\) is \(\tan \alpha + C\), where \(C\) is the constant of integration. Thus,\[\int \sec^2 \alpha \, d\alpha = \tan \alpha + C\].

Step 4: State the Final Answer

Combine the results from the previous steps. The general indefinite integral of \(\int (1 + \tan^2 \alpha) \, d\alpha\) is:\[\tan \alpha + C\].

Key Concepts

Pythagorean Identities

Pythagorean Identities are fundamental relationships in trigonometry that relate the squares of the sine, cosine, and tangent functions. In this exercise, the identity used is \(1 + \tan^2 \alpha = \sec^2 \alpha\). This particular identity is derived from the well-known Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), which forms the basis for several other identities.
To derive \(1 + \tan^2 \alpha = \sec^2 \alpha\), remember that \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\sec \alpha = \frac{1}{\cos \alpha}\). By manipulating the basic Pythagorean identity and substituting these expressions, we can reach the trigonometric identity used in the solution.
Understanding these identities is crucial, as they simplify integrals by providing equivalent expressions that are often easier to integrate.

Trigonometric Functions

Trigonometric Functions, like sine, cosine, tangent, and secant, are the building blocks of trigonometry. They model periodic phenomena such as sound waves and light. In our exercise, the functions we focus on are tangent, \( \tan \alpha \), and secant, \( \sec \alpha \).
These functions have specific properties and derivatives which play important roles in calculus, particularly in integration.

• The tangent function, \( \tan \alpha \), represents the ratio of the opposite side to the adjacent side in a right-angled triangle.
• The secant function, \( \sec \alpha \), is the reciprocal of the cosine function, defined as \( \frac{1}{\cos \alpha} \).

Understanding these functions and their relationships through identities allows us to transform expressions, making them suitable for techniques like integration.

Integration Techniques

Integration Techniques are methods used to find integrals, which are the reverse operation of differentiation. In the step-by-step solution, we used a substitution from the Pythagorean identities to simplify the integral. This is a common integration technique that makes use of known identities to rewrite the integral in a more recognizable form.

• Firstly, remember to identify any familiar patterns or identities, like \(1 + \tan^2 \alpha = \sec^2 \alpha\), which simplify the integrand.
• After substituting, integrating \(\sec^2 \alpha\) becomes straightforward since its antiderivative is a well-known function, \(\tan \alpha + C\).

Mastering different integration techniques, such as substitution and recognizing identities, is essential for solving more complex integrals in calculus. These skills enhance problem-solving efficiency by reducing the complexity of the expressions you need to integrate.