Problem 39 - 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. $$\int\left(-3 \csc ^{2} x\right) d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \(3\cot x + C\).

Step 1: Identify the Function to Integrate

Given the integral \( \int -3 \csc^2 x \, dx \), identify the main function. Here, the function is \(-3 \csc^2 x\), which is a direct trigonometric identity related to derivatives.

Step 2: Recall Known Integral Formulas

Recall that the derivative of \(-\cot x\) is \(-\csc^2 x\). This suggests that the integral of \(-\csc^2 x\) is \(\cot x\), potentially multiplied by a constant if present in the expression.

Step 3: Apply the Integral Formula

Apply the integral formula by integrating \(-3 \csc^2 x\) directly: \[ \int -3 \csc^2 x \, dx = -3 \cdot (\int \csc^2 x \, dx) = -3 (-\cot x) + C \], which simplifies to \(3\cot x + C\), where \(C\) is the constant of integration.

Step 4: Differentiate to Verify

Differentiate the proposed antiderivative \(3 \cot x + C\) to confirm it matches the original function. The derivative of \(3 \cot x\) is \(-3 \csc^2 x\), and since the derivative of a constant is zero, it confirms our solution is correct as it matches the original integrand.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve integrating functions that contain trigonometric functions like sine, cosine, tangent, and their reciprocals. These types of integrals are common in calculus because trigonometric functions frequently appear in physics, engineering, and mathematics.
In this exercise, integrating the function \(-3\csc^2 x\) offers practice in handling trigonometric integrals. The core idea is to

• Recognize the trigonometric identity or derivative involved, which often guides you to the antiderivative.
• Recall known derivative and integral pairs like \(\int \csc^2 x\, dx = -\cot x\).
• Adjust for constants multiplying the integrals, as seen by the -3 factor in our case.

These steps are crucial since recognizing the pattern simplifies the process of finding the antiderivative. Remember that verification through differentiation is a key step in ensuring our solution matches the original function.

Indefinite Integral

An indefinite integral refers to the antiderivative of a function and includes a constant of integration, \(C\). Unlike definite integrals that calculate area under a curve between two bounds, indefinite integrals find the most general form of a function from its rate of change.
When we talk about the indefinite integral of a trigonometric function like \(-3\csc^2 x\), our goal is to determine what function, when differentiated, gives back the original expression. This leads us to the antiderivative. In this case:

• The antiderivative of \(-\csc^2 x\) is \(\cot x\).
• Multiply by any constants present, as with the -3 factor.
• Always add the constant \(C\) at the end of the integration.

This process results in \(3 \cot x + C\), capturing all possible functions whose derivative yields \(-3 \csc^2 x\).

Integration Techniques

Integration techniques in calculus are varied and provide different approaches to tackle functions you need to integrate. The technique used may depend on the type of function. For trigonometric integrals, like the one in our exercise, recognizing patterns is vital.
Key techniques for handling

• Utilization of known integral formulas that directly relate to derivatives of trigonometric functions.
• Adjustment for coefficients like the constant -3 here.
• Verification by differentiation, which confirms the correctness of the integration performed.

These techniques make integration manageable and ensure accuracy. By simplifying complex expressions and using these strategies, solving integrals becomes more intuitive. The ability to switch between recognizing derivatives and recalling integral formulas is essential in mastering trigonometric integrals.