Problem 74 - University Calculus: Early Transcendentals

Question

Verify the formulas in Exercises by differentiation. $$\int \csc ^{2}\left(\frac{x-1}{3}\right) d x=-3 \cot \left(\frac{x-1}{3}\right)+C$$

Step-by-Step Solution

Answer

The formula is verified as correct.

Step 1: Differentiate the Right-Hand Side

To verify the integral, differentiate the given expression on the right-hand side: -3 \cot \left(\frac{x-1}{3}\right) + C. To find the derivative, use the chain rule. The derivative of \( \cot(u) \) with respect to \( u \) is \( -\csc^2(u) \). Here, \( u = \frac{x-1}{3} \), so we also need to differentiate \( \frac{x-1}{3} \) with respect to \( x \), which is \( \frac{1}{3} \).

Step 2: Apply the Chain Rule

Apply the chain rule to differentiate:\[ \frac{d}{dx}[-3 \cot \left(\frac{x-1}{3}\right)] = -3 \left( -\csc^2 \left(\frac{x-1}{3}\right) \right) \cdot \left( \frac{1}{3} \right) \]This simplifies to:\[ \csc^2 \left(\frac{x-1}{3}\right) \]

Step 3: Simplify the Result

Simplify the differentiation result from Step 2:The differentiation yields \( \csc^2 \left(\frac{x-1}{3}\right) \), which matches the original integrand of the integral. This confirms that the integration was performed correctly.

Step 4: Conclusion

Since the differentiation of \( -3 \cot \left(\frac{x-1}{3}\right) + C \) gives us \( \csc^2 \left(\frac{x-1}{3}\right) \), we have verified the formula is correct. The integral's antiderivative matches the given function.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function. The derivative represents the rate at which a function changes at any given point. In simple terms, it tells us how the function is "sloping" at any moment.

• The derivative of a constant is zero.
• For power functions, use the power rule: if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• For trigonometric functions such as \( \cot(x) \), the derivative is \( -\csc^2(x) \).

In the exercise, we begin by differentiating the right-hand side of the given equation. By knowing the rules of differentiation, you are equipped to find whether this operation accurately leads back to the original integrand.

Chain Rule

The chain rule is a technique used in calculus to differentiate composite functions. It allows you to understand how functions nested inside each other change. The formula of the chain rule is:\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
When a function is composed of an outer function and an inner function, you use the chain rule to differentiate effectively. First, you differentiate the outer function while keeping the inner function the same. Then, multiply this by the derivative of the inner function.
In the provided solution, the chain rule was applied to \(-3 \cot\left(\frac{x-1}{3}\right)\). Here, the outer function was \(-3 \cot(u)\) where \(u = \frac{x-1}{3}\). After differentiating \(-3 \cot(u)\) into \(3 \csc^2(u)\) and multiplying by \(\frac{1}{3}\) — the derivative of the inner function — we achieve the goal of verifying differentiation with the given function.

Antiderivative

An antiderivative, also known as an indefinite integral, is essentially the reverse process of differentiation. It aims to find a function whose derivative will result in the original function.

• The process involves integrating the given function.
• Finding an antiderivative is not always straightforward, especially for complex functions.
• Each antiderivative includes a constant \(C\), accounting for the fact that multiple functions can have the same derivative.

In the context of this exercise, you are given the integral of \( \csc^2\left(\frac{x-1}{3}\right) \) which results in \(-3 \cot\left(\frac{x-1}{3}\right) + C\). Differentiating this outcome unravels the solution back to the original integrand, confirming a successful integration.

Trigonometric Integrals

Trigonometric integrals involve integration of functions that blend trigonometric functions. These require special strategies because of the cyclical nature of trigonometric functions.

• Common techniques involve using trigonometric identities or substitution methods.
• Explicit formulas exist for specific functions, such as \( \int \csc^2(x)dx = -\cot(x) + C \).
• Trigonometric integrals can often be simplified using known antiderivatives.

In this exercise, the integral of the trigonometric function \( \csc^2\left(\frac{x-1}{3}\right) \) leads to an antiderivative involving \(-\cot\left(\frac{x-1}{3}\right)\). Recognizing and applying these patterns aids in correctly performing integrations.