Problem 51 - Thomas Calculus

Question

In Exercises \(17-56,\) 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. $$\begin{array}{l}{\int\left(1+\tan ^{2} \theta\right) d \theta} \\\ {\left(\operatorname{Hint} : 1+\tan ^{2} \theta=\sec ^{2} \theta\right)}\end{array}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

We are asked to find the antiderivative of \( \int(1 + \tan^2 \theta) \, d\theta \). The hint given is that \( 1 + \tan^2 \theta = \sec^2 \theta \).

Step 2: Substitute the Hint

Using the hint, substitute \( \sec^2 \theta \) for \( 1 + \tan^2 \theta \) in the integral. Thus, the integral becomes \( \int \sec^2 \theta \, d\theta \).

Step 3: Antiderivative of \( \sec^2 \theta \)

Recall that the antiderivative of \( \sec^2 \theta \) is \( \tan \theta + C \), where \( C \) is the constant of integration.

Step 4: Conclude the Antiderivative

Therefore, the most general antiderivative is \( \tan \theta + C \).

Step 5: Verification by Differentiation

Differentiate \( \tan \theta + C \). The derivative of \( \tan \theta \) is \( \sec^2 \theta \), and since \( C \) is a constant, its derivative is 0. Therefore, the derivative of \( \tan \theta + C \) is \( \sec^2 \theta \), confirming our antiderivative is correct.

Key Concepts

Antiderivative

Finding an antiderivative involves reversing the process of differentiation. It's like "undoing" a derivative to get back to the original function. In calculus, the antiderivative is often referred to as the indefinite integral, which differs from definite integrals that calculate a specific area under a curve. In the exercise, \[\int(1 + \tan^2 \theta) \, d\theta\]is an indefinite integral, because it doesn't have upper and lower limits of integration.
To solve for the antiderivative, remember that it refers to a family of functions whose derivative yields the integrand. This concept is crucial, as it means there can be multiple antiderivatives differing by a constant of integration. In cases like the exercise, where you have a trigonometric identity simplification, it's easier to find the antiderivative by following the mathematical transformations given, leading you to the solution quickly.

Trigonometric Identities

In the realm of calculus, trigonometric identities are like secret tools that can simplify complex expressions. Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. In this exercise, we use the identity:
\[1 + \tan^2 \theta = \sec^2 \theta\]
This is known as a Pythagorean identity, and it transforms the given integral into something easier to work with.
By substituting \(\sec^2 \theta\) into the integral \[\int(1 + \tan^2 \theta) \, d\theta\]we make the expression simpler and more straightforward to integrate.
Trigonometric identities are powerful because they can change a difficult integral into one that is much easier to solve. Always keep these handy to look for opportunities to substitute and simplify wherever possible.

Constant of Integration

In calculus, a very important component of indefinite integrals is the constant of integration, represented as \(C\). Whenever you find an indefinite integral, you're essentially looking for a family of functions, all of which derive to the same integrand. This is where \(C\) becomes crucial.
In the exercise, the solution \[\int \sec^2 \theta \, d\theta = \tan \theta + C\]contains this constant. It accounts for the fact that there isn't just one function with a given derivative, but rather an infinite family, each differing by a constant.
The constant of integration reminds us that while the derivative of a constant is zero, the definite and precise function that served as the antiderivative before differentiation could include any constant value. Hence, every indefinite integral comes with this constant, which signifies the sheer family of functions as solutions.

Differentiation Verification

Verification through differentiation is a powerful technique to confirm that you've found the correct antiderivative. This method involves deriving the obtained antiderivative to check if you land on the original integrand, essentially backtracking your steps.
For the integral in question, the antiderivative \(\tan \theta + C\) is derived in this way. Differentiating \(\tan \theta\) gives \(\sec^2 \theta\), and the derivative of a constant \(C\) is 0. Therefore,\[\frac{d}{d\theta}(\tan \theta + C) = \sec^2 \theta\]
This matches exactly with the transformed integrand, confirming our solution is correct.
Verification by differentiation provides a robust check and is indispensable for ensuring the solution to an indefinite integral is accurate. It makes the problem-solving process precise and complete.