Problem 40 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int\left(-\frac{\sec ^{2} x}{3}\right) d x $$

Step-by-Step Solution

Answer

The most general antiderivative is \( -\frac{1}{3} \tan x + C \).

Step 1: Identify the Antiderivative Formula

The integral we need to evaluate is \[\int \left( -\frac{\sec^2 x}{3} \right) \, dx.\]We know from calculus that the derivative of \( \tan x \) is \( \sec^2 x \). Therefore, the antiderivative of \( \sec^2 x \) is \( \tan x \).

Step 2: Extract Constants From the Integral

We can take constants out of the integral:\[-\frac{1}{3} \int \sec^2 x \, dx.\]

Step 3: Find the Antiderivative

Now we can integrate:\[-\frac{1}{3} \int \sec^2 x \, dx = -\frac{1}{3} \tan x + C.\]Here, \( C \) is the constant of integration.

Step 4: Differentiate to Verify the Solution

Differentiate our result to ensure correctness. The derivative of \( -\frac{1}{3} \tan x + C \) is:\[-\frac{1}{3} \sec^2 x\]which matches the original integrand \( -\frac{\sec^2 x}{3} \). This confirms the antiderivative is correct.

Key Concepts

Antiderivatives

Antiderivatives are fundamental concepts in calculus. They refer to the function you get when reversing differentiation. For example, if differentiating a function gives you a particular result, the reverse process would give you the original function back.

To find the antiderivative of a given function, one looks for a function whose derivative matches the function in question. This process is often represented with the integral symbol \( \int \). An integral that doesn't have specific bounds is called an 'indefinite integral.'

In our example, we need to find the antiderivative of \( -\frac{1}{3} \sec^2 x \). From calculus, we know that the derivative of \( \tan x \) is \( \sec^2 x \), so the antiderivative of \( \sec^2 x \) is \( \tan x \). Thus, the antiderivative of the given function is:- \(-\frac{1}{3} \tan x\).
Finding antiderivatives can sometimes require recognizing standard derivative formulas or using integration techniques.

Constant of Integration

When evaluating indefinite integrals, it is essential to include the constant of integration, denoted by \( C \).

The reason for this is that when we differentiate, the derivative of a constant is zero. Therefore, several different functions can have the same derivative if they differ only by a constant. To address this in antiderivatives, we always add a constant of integration.

In our example, after finding the antiderivative of \( \int \sec^2 x \, dx \), we added \( C \). Thus, our complete solution to the integral \( \int \left( -\frac{\sec^2 x}{3} \right) dx \) is \(-\frac{1}{3} \tan x + C\).
Remember:- The presence of \( C \) indicates there are actually infinitely many antiderivatives, each differing by a constant.

Differentiation Verification

Verification by differentiation is a powerful tool to ensure the accuracy of finding an antiderivative. To verify, you simply differentiate the antiderivative. If you end up with the original integrand, your solution is correct.

In the exercise at hand, after computing that the antiderivative was \(-\frac{1}{3} \tan x + C\), we differentiated it:- the derivative of \(-\frac{1}{3} \tan x\) yields \(-\frac{1}{3} \sec^2 x\).

This matches the original expression within the integral, \(-\frac{\sec^2 x}{3}\), confirming the antiderivative is accurate. This step is crucial to cement understanding and correct any mistakes made during integration.
Differentiation verification:- Helps confirm the correctness of the antiderivative- Provides confidence in integration results