Problem 52 - Calculus Single Variable

Question

Evaluate the given integral by converting the integrand to an expression in sines and cosines. $$ \int 4 \cot (x / 2) \csc ^{4}(x / 2) d x $$

Step-by-Step Solution

Answer

\( \int 4 \cot(x/2) \csc^4(x/2) \, dx = -2 \csc^4(x/2) + C. \)

Step 1: Identify Trigonometric Identities

To evaluate the integral \( \int 4 \cot(x/2) \csc^4(x/2) \, dx \), first express the trigonometric functions \( \cot(x/2) \) and \( \csc(x/2) \) in terms of sine and cosine. Recall that \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Step 2: Rewrite the Integrand

Substitute \( \cot(x/2) \) and \( \csc(x/2) \) using their sine and cosine equivalents: \[ 4 \cot(x/2) \csc^4(x/2) = 4 \left( \frac{\cos(x/2)}{\sin(x/2)} \right) \left( \frac{1}{\sin^4(x/2)} \right) = \frac{4 \cos(x/2)}{\sin^5(x/2)}. \]

Step 3: Simplify the Integrand

Combine the fractions in the expression: \[ \frac{4 \cos(x/2)}{\sin^5(x/2)} = 4 \cos(x/2) \sin^{-5}(x/2). \]

Step 4: Make a Substitution

Let \( u = \sin(x/2) \), then \( du = \frac{1}{2} \cos(x/2) \ dx \), or equivalently \( 2\, du = \cos(x/2) \ dx \). Thus, the integrand becomes: \[ \int 4 \cos(x/2) \sin^{-5}(x/2) \, dx = \int 8 u^{-5} \, du. \]

Step 5: Integrate with Respect to \( u \)

To integrate \( 8 u^{-5} \), use the power rule for integration: \[ \int 8 u^{-5} \, du = 8 \int u^{-5} \, du = 8 \left( \frac{u^{-4}}{-4} \right) = -2 u^{-4} + C. \]

Step 6: Substitute Back for \( x \)

Replace \( u = \sin(x/2) \) back into the expression: \[ -2 u^{-4} + C = -2 \sin^{-4}(x/2) + C. \]

Step 7: Final Expression

Thus, the evaluated integral is: \[ \int 4 \cot(x/2) \csc^4(x/2) \, dx = -2 \csc^4(x/2) + C. \]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, especially when dealing with complex integrals. They allow us to convert expressions involving tangent, cotangent, secant, and cosecant into simpler terms using sine and cosine. For example, we know that \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and \( \csc(\theta) = \frac{1}{\sin(\theta)} \). These relationships are essential for simplifying integrals. By expressing an integrand like \( 4 \cot(x/2) \csc^4(x/2) \) in terms of sine and cosine, calculations become straightforward. This approach helps in integrating trigonometric expressions efficiently by reducing them to forms where standard techniques can be applied.

Trigonometric Substitution

Trigonometric substitution is a powerful integration technique used to simplify integrals involving trigonometric expressions. By using identities and substitutions, we can transform an integrand into a different variable, making it easier to integrate. In our problem, we perform a trigonometric substitution by letting \( u = \sin(x/2) \). When this substitution is applied, it simplifies the integration process. By changing variables, the integral \( \int 4 \cos(x/2) \sin^{-5}(x/2) \, dx \) turns into \( \int 8 u^{-5} \, du \). This process highlights the utility of trigonometric substitution in converting a difficult integral into a more manageable form, leveraging the relationships between trigonometric functions and their derivatives.

Integration Techniques

In calculus, integration techniques are methods to find the integral of complex functions. Techniques like substitution and integration by parts allow us to integrate expressions that aren't immediately solvable. In this exercise, substitution simplifies the integral greatly. Recognize how each step systematically builds upon the results of trigonometric substitutions. We changed both the integrand and the variable of integration to leverage a simpler expression \( \int 8 u^{-5} \, du \). From here, we used the power rule for integration: \( \, \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \, \), to integrate easily, resulting in \( -2 u^{-4} + C \). This showcases how strategic choices of techniques simplify integration problems.

Definite and Indefinite Integrals

Integrals come in two flavors: definite and indefinite. Indefinite integrals, like the one in our exercise, represent a family of antiderivatives and include a constant of integration \( C \). This is apparent when we find that \( \int 4 \cot(x/2) \csc^4(x/2) \, dx = -2 \csc^4(x/2) + C \). Definite integrals, in contrast, compute the area under a curve over a specific interval. Recognizing the nature of the integral type is important, as it defines the approach and purpose. In teaching integral calculus, emphatically differentiating these types helps contextualize when a constant of integration is necessary, versus computing a specific value, characteristic of definite integrals.