Problem 57 - University Calculus: Early Transcendentals

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

Step-by-Step Solution

Answer

The most general antiderivative is \(-\frac{1}{2}\cos 2x + \cot x + C\).

Step 1: Understand the integral

We need to find the indefinite integral of \( \int (\sin 2x - \csc^2 x) \ dx \). This means we should find a function \( F(x) \) such that \( F'(x) = \sin 2x - \csc^2 x \).

Step 2: Break Down the Integral

The integral \( \int (\sin 2x - \csc^2 x) \ dx \) can be split into two separate integrals: \( \int \sin 2x \ dx - \int \csc^2 x \ dx \).

Step 3: Integrate \( \sin 2x \)

Use a substitution: Let \( u = 2x \), so \( du = 2 \, dx \) or \( dx = \frac{1}{2} du \). This transforms the integral into \( \frac{1}{2} \int \sin u \ du \). The antiderivative of \( \sin u \) is \( - \cos u \), so we have:\[-\frac{1}{2} \cos u + C_1 = -\frac{1}{2} \cos (2x) + C_1.\]

Step 4: Integrate \( \csc^2 x \)

The integral of \( \csc^2 x \) is a standard result: \( \int \csc^2 x \, dx = -\cot x + C_2 \).

Step 5: Combine the Results

Combine the results from Step 3 and Step 4: \[ \int (\sin 2x - \csc^2 x) \ dx = \left(-\frac{1}{2} \cos 2x + C_1\right) + \left(\cot x + C_2\right). \] Combine the constants \( C_1 \) and \( C_2 \) into a single constant \( C \):\[-\frac{1}{2} \cos 2x + \cot x + C.\]

Step 6: Verify the Solution

Differentiate the result to ensure we return to the original integrand:1. Derivative of \(-\frac{1}{2} \cos 2x\): \( \frac{1}{2} \cdot 2 \cdot \sin 2x = \sin 2x \).2. Derivative of \( \cot x \): \(-\csc^2 x \).Combine: \( \sin 2x - \csc^2 x \), which matches the original integrand.

Key Concepts

Integration Techniques

Finding an indefinite integral involves using various integration techniques to determine the antiderivative of a given function. In this exercise, the function \((\sin 2x - \csc^2 x)\) is broken down into simpler parts for easier integration. When dealing with complex expressions, breaking them into simpler integrals is often the key.Some common integration techniques include:

• Substitution: Transform variables to make integration simpler.
• Integration by Parts: A method useful when the integrand is a product of functions.
• Partial Fraction Decomposition: Useful for rational functions.

In the original exercise, substitution was used to handle \(\sin 2x\), by letting \(u = 2x\) which simplifies the integration process. Understanding these methods serves as a toolkit that helps in finding antiderivatives of complex functions.

Trigonometric Integrals

Trigonometric integrals involve functions that include trigonometric expressions like sine, cosine, and their reciprocals. In this exercise, the functions \(\sin 2x\) and \(\csc^2 x\) are typical trigonometric expressions that we often encounter.Key points about trigonometric integrals:

• They often require the use of trig identities to simplify the integrals.
• Common trigonometric integrals, like \(\int \sin x \, dx = -\cos x + C\), should be memorized for quick access.
• Substitution is common, especially when angles are multiples (e.g., using \(u = 2x\) for \(\sin 2x\)).

For \(\csc^2 x\), it's a known standard integral with the result \(-\cot x\). By thoroughly knowing these standard results, integrating related functions becomes much more manageable.

Antiderivatives

Antiderivatives, or indefinite integrals, are functions that reverse differentiation. Given a function, finding its antiderivative means determining a new function whose derivative yields the original function. In the context of this exercise, we are tasked with reversing the differentiation of \(\sin 2x - \csc^2 x\).Important aspects of antiderivatives:

• Antiderivatives are not unique. They have an arbitrary constant \(C\) added, representing the family of all functions whose derivative gives the specific form.
• Differentiating the antiderivative should yield the original function to verify correctness.
• This exercise combined constants \(C_1\) and \(C_2\) into one \(C\) to express the most general solution.

By understanding which operations reverse differentiation, such as integrating basic trigonometric functions, you can find antiderivatives of more complex expressions effectively.