Problem 51 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \begin{array}{l}{\int \cot ^{2} x d x} \\ {\left(\text {Hint} : 1+\cot ^{2} x=\csc ^{2} x\right)}\end{array} $$

Step-by-Step Solution

Answer

The most general antiderivative is \(-\cot x - x + C\).

Step 1: Use Trigonometric Identity

To find the antiderivative of \( \cot^2 x \), use the hint provided in the problem: \(1 + \cot^2 x = \csc^2 x\). This can be rewritten to express \( \cot^2 x \) as \( \csc^2 x - 1 \).

Step 2: Rewrite the Integral

Substitute \( \cot^2 x \) with \( \csc^2 x - 1 \) in the integral: \[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx. \]

Step 3: Separate the Integral

Separate the terms inside the integral: \[ \int (\csc^2 x) \, dx - \int 1 \, dx. \]

Step 4: Find the Antiderivatives

Find the antiderivatives of each term. The antiderivative of \( \csc^2 x \) is \( -\cot x \), and the antiderivative of \( 1 \) is \( x \). Thus, \[ \int \csc^2 x \, dx = -\cot x \] and \[ \int 1 \, dx = x. \]

Step 5: Combine the Antiderivatives

Combine the antiderivatives and include the constant of integration, \( C \): \[ \int (\csc^2 x - 1) \, dx = -\cot x - x + C. \]

Step 6: Verify by Differentiation

To check the result, differentiate \(-\cot x - x + C\) and see if it equals \(\cot^2 x \). The derivative of \(-\cot x\) is \( \csc^2 x \), and the derivative of \(-x\) is \(-1\), so: \[ \frac{d}{dx} (-\cot x - x + C) = \csc^2 x - 1 = \cot^2 x. \]This confirms our solution is correct.

Key Concepts

Antiderivative

The concept of an antiderivative is fundamental in calculus. It can be thought of as the reverse of taking a derivative. While a derivative represents the rate of change of a function, an antiderivative takes us back to the original function. Finding an antiderivative involves determining a function whose derivative is the given function.It's important to note that there is always an infinite number of antiderivatives due to the constant of integration, represented as "C." This is because, when differentiating, the constant disappears. For example, if the derivative of a function is a constant, say 3, then the antiderivative could be multiple possibilities such as 3x, 3x + 1, 3x - 7, and so on. All of these have the same derivative, which is the rule showing this multiplicity.

Indefinite Integral

The indefinite integral, sometimes known as an antiderivative, is used to denote the whole family of antiderivatives of a function. It is represented by the integral symbol \( \int \), followed by the function and the differential \( dx \). This notation communicates that we're seeking an antiderivative rather than a specific value, which is the case with definite integrals.When you're working with indefinite integrals:

• No limits of integration are specified, meaning the result is a general expression plus a constant of integration, "C."
• It's crucial to apply integration techniques, like substitution, to simplify the problem effectively.
• As with derivatives, understanding the rules and properties of indefinite integrals is essential for solving complex calculations.

In the exercise, finding \( \int \cot^2 x \, dx \) involved rewriting using trigonometric identities and separating terms to apply integration rules individually.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the identity are defined. They are extremely useful in simplifying integrals and differential equations involving trigonometric functions.Some common trigonometric identities include:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \cot^2 x = \csc^2 x \)
• \( 1 + \tan^2 x = \sec^2 x \)

In calculus, such identities help in rewriting expressions to make them more manageable for integration or differentiation. For instance, in the given problem, using \( 1 + \cot^2 x = \csc^2 x \) allowed us to rewrite the integral in a form where standard integration techniques were straightforwardly applicable. This reveals how identities can simplify seemingly complex mathematical problems.