Problem 69 - Thomas Calculus

Question

Use reduction formulas to evaluate the integrals. \(\int \csc ^{5} x d x\)

Step-by-Step Solution

Answer

The integral is: \( -\frac{1}{4} \csc^{3}(x) \cot(x) -\frac{3}{8} \csc(x) \cot(x) + \frac{3}{8} \ln |\csc(x) - \cot(x)| + C \).

Step 1: Understand the Reduction Formula

The reduction formula for the integral of \( \csc^n(x) \) is given by:\[\int \csc^n(x) \, dx = -\frac{1}{n-1} \csc^{n-2}(x) \cot(x) + \frac{n-2}{n-1} \int \csc^{n-2}(x) \, dx\]This helps simplify integrals of the form \( \csc^n(x) \) to a previously solved case.

Step 2: Apply the Reduction Formula to \( \csc^5(x) \)

We start with \( n = 5 \) in the reduction formula:\[\int \csc^5(x) \, dx = -\frac{1}{4} \csc^{3}(x) \cot(x) + \frac{3}{4} \int \csc^{3}(x) \, dx\]This step reduces the problem to evaluating \( \int \csc^{3}(x) \, dx \).

Step 3: Reduce Further with \( \csc^3(x) \)

Applying the reduction formula again for \( n = 3 \):\[\int \csc^3(x) \, dx = -\frac{1}{2} \csc(x) \cot(x) + \frac{1}{2} \int \csc(x) \, dx\]The integral \( \int \csc(x) \, dx \) is known, allowing us to solve.

Step 4: Solve Known Integral \( \int \csc(x) \, dx \)

The integral \( \int \csc(x) \, dx \) is:\[\int \csc(x) \, dx = \ln |\csc(x) - \cot(x)| + C\]This result will be used in completing the solution.

Step 5: Simplify and Substitute Back

Substitute the known integral back into the equation from Step 3:\[\int \csc^3(x) \, dx = -\frac{1}{2} \csc(x) \cot(x) + \frac{1}{2} \left( \ln |\csc(x) - \cot(x)| \right) + C_1\]Insert this into the expression from Step 2 to fully solve \( \int \csc^5(x) \, dx \).

Step 6: Final Integration Result

Combine all elements for the original integral:\[\int \csc^5(x) \, dx = -\frac{1}{4} \csc^{3}(x) \cot(x) + \frac{3}{4} \left( -\frac{1}{2} \csc(x) \cot(x) + \frac{1}{2} \ln |\csc(x) - \cot(x)| \right) + C\]Simplify to:\[\int \csc^5(x) \, dx = -\frac{1}{4} \csc^{3}(x) \cot(x) -\frac{3}{8} \csc(x) \cot(x) + \frac{3}{8} \ln |\csc(x) - \cot(x)| + C\]

Key Concepts

Cosecant Function Integration

Integrating the cosecant function can be a bit tricky, but it's a crucial part of understanding many trigonometric integrals. The cosecant function, expressed as \( \csc(x) = \frac{1}{\sin(x)} \), often appears in integrals that require specific techniques for solving. Understanding how to integrate expressions like \( \int \csc(x) \, dx \) helps in tackling more complex integrals, such as higher powers of cosecant. Typically, these integrals involve consideration of other functions like cotangent, as we often see cosecant paired with cotangent identities. To solve cosecant integrals, it is helpful to use logarithmic identities, especially since the integral \( \int \csc(x) \, dx \) results in \( \ln |\csc(x) - \cot(x)| + C \). Mastering this basic integral is essential as it serves as a building block for dealing with more complicated trigonometric integrals.

Reduction Formula for Integrals

Reduction formulas are powerful tools in calculus, helping us solve complex integrals by reducing them to simpler forms. The concept of a reduction formula involves expressing an integral in terms of another integral with a lower power or simpler form. For trigonometric functions like cosecant, a reduction formula simplifies the integration process by connecting the integral of \( \csc^n(x) \) with the integral of \( \csc^{n-2}(x) \). For instance, the formula for \( \int \csc^n(x) \, dx \) is:

• \(-\frac{1}{n-1} \csc^{n-2}(x) \cot(x)\)
• \(+ \frac{n-2}{n-1} \int \csc^{n-2}(x) \, dx \)

This sequential reduction allows us to break down seemingly unmanageable integrals into those that we can integrate directly or relate to known results, like \( \int \csc(x) \, dx \). Learning to apply reduction formulas effectively streamlines the solving process for complex trigonometric integrals.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions that include trigonometric identities—sine, cosine, tangent, cosecant, secant, and cotangent. These integrals are paramount in solving numerous calculus problems, particularly in physics and engineering. Often, trigonometric integrals require a mix of strategies, such as applying identities, substitution, or integration by parts. In the context of cosecant integrals, often, we have to deal with combinations of trigonometric functions, bringing in cotangent functions as well. A solid grasp of trigonometric identities, like \( \csc^2(x) = \cot^2(x) + 1 \) or the Pythagorean identities, greatly aids in integrating these functions. Approaching trigonometric integrals typically involves:

• Simplifying the integrand using identities.
• Applying reduction formulas or substitution.
• Breaking down difficult integrals into simpler, known integrals.

Understanding these methods makes it easier to navigate the often complicated landscapes of trigonometric integrals.

Integration Techniques

Integration techniques are diverse and crucial to solving a wide array of mathematical problems, particularly those that involve integrating functions not easily handled by basic methods. These techniques include but are not limited to:

• Substitution: Simplifies integrands by changing variables.
• Integration by Parts: Useful when dealing with products of functions, using the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fractions: Breaks down complex rational functions into simpler fractions.
• Trigonometric Substitution: Deals with integrands involving \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).

When tackling the integral of a function like \( \csc^5(x) \), using specific techniques like reduction formulas allows for systematic integration, reducing the complexity step by step. Mastery of these techniques equips students to solve integrals that may initially seem daunting, turning them into manageable challenges with a clear pathway to the solution.