Problem 51 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\begin{aligned} &\int\left(1+\tan ^{2} \theta\right) d \theta\\\ &\text { (Hint: }\left.1+\tan ^{2} \theta=\sec ^{2} \theta\right) \end{aligned}$$

Step-by-Step Solution

Answer

The most general antiderivative is \(\tan \theta + C\).

Step 1: Recognize the trigonometric identity

The integral to solve is \(\int (1+\tan^2 \theta) \, d\theta\). Use the hint provided: \(1+\tan^2 \theta = \sec^2 \theta\), to rewrite the integral as \(\int \sec^2 \theta \, d\theta\).

Step 2: Recall the antiderivative formula

The next step is to recognize the formula for the antiderivative of \(\sec^2 \theta\), which is a standard result in calculus: \(\int \sec^2 \theta \, d\theta = \tan \theta + C\), where \(C\) is the integration constant.

Step 3: Solve the integral

Using the result from Step 2, we solve the integral: \(\int \sec^2 \theta \, d\theta = \tan \theta + C\). Thus, the most general antiderivative of \(1+\tan^2 \theta\) is \(\tan \theta + C\).

Step 4: Verify by differentiation

Differentiate the antiderivative \(\tan \theta + C\) to ensure it equals the original integrand. The derivative of \(\tan \theta\) is \(\sec^2 \theta\), and the derivative of a constant \(C\) is 0. Therefore, \(\frac{d}{d\theta}(\tan \theta + C) = \sec^2 \theta\), which matches the rewritten integrand, confirming the antiderivative is correct.

Key Concepts

Antiderivatives

Antiderivatives are a key concept in calculus, often referred to as the reverse process of differentiation. When you find an antiderivative of a function, you are essentially finding a function whose derivative is the original function. This process is helpful when solving various problems in calculus and physics.

In our exercise, we needed to identify the antiderivative of the function involving trigonometric terms: \(1 + \tan^2 \theta\). First, we used a trigonometric identity to rewrite the integral as \(\int \sec^2 \theta \, d\theta\). Recognizing antiderivatives can often require remembering or looking up common integral and derivative pairs.

• The antiderivative of \(\sec^2 \theta\) is \(\tan \theta + C\), a standard result in calculus.
• The "+ C" term is essential and represents the constant of integration, a crucial part of indefinite integrals.

Understanding antiderivatives helps solve many practical problems, such as calculating areas and solving differential equations.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable involved. They are fundamental tools in calculus, particularly when dealing with integrals and derivatives of trigonometric functions.

In our exercise, we applied the identity \(1 + \tan^2 \theta = \sec^2 \theta\). This identity is crucially important as it simplifies the integral considerably, making it more straightforward to find the antiderivative.

• Knowing common identities allows for simplification in mathematical expressions.
• They are especially useful for integrating or differentiating when the function does not appear in a standard form.

By transforming expressions using identities, we can often find solutions with less effort, which also verifies the connection between trigonometric functions.

Indefinite integrals

Indefinite integrals, also known as antiderivatives, represent a family of functions and are central in the study of calculus. Unlike a definite integral, which calculates the area under a curve between two points, an indefinite integral represents a general form of the antiderivative for a function.

Our exercise involved solving the indefinite integral \(\int (1 + \tan^2 \theta) \, d\theta\). By applying trigonometric identities, we rewrote it simply as \(\int \sec^2 \theta \, d\theta\) and subsequently found its antiderivative as \(\tan \theta + C\).

• Indefinite integrals are useful for finding general solutions in differential equations.
• The integration constant "C" accounts for any constant term that would disappear when differentiating back.

Understanding indefinite integrals leads to discovering functions that can model real-world scenarios, facilitating predictions and deeper insights into dynamic systems.