Problem 53 - Calculus Single Variable

Question

Evaluate the given integral by converting the integrand to an expression in sines and cosines. $$ \int \cot ^{3}(x / 3) \csc (x / 3) d x $$

Step-by-Step Solution

Answer

\(-\csc^3(x/3) + 3\csc(x/3) + C\)

Step 1: Express the cotangent and cosecant in terms of sine and cosine.

The cotangent and cosecant functions can be rewritten in terms of sine and cosine as follows:\[ \cot(u) = \frac{\cos(u)}{\sin(u)} \quad \text{and} \quad \csc(u) = \frac{1}{\sin(u)} \]Thus, the integrand \( \cot^{3}(x / 3) \csc(x / 3) \) becomes:\[ \left(\frac{\cos(x/3)}{\sin(x/3)}\right)^3 \cdot \frac{1}{\sin(x/3)} = \frac{\cos^{3}(x/3)}{\sin^{4}(x/3)} \]

Step 2: Simplify the integrand using a trigonometric substitution.

Let \( u = \sin(x/3) \), then \( du = \frac{1}{3} \cos(x/3) \, dx \) which implies \( dx = 3 \frac{du}{\cos(x/3)} \). Substitute these into the integrand:\[ \int \frac{\cos^{3}(x/3)}{\sin^{4}(x/3)} \, dx = \int \frac{\cos(x/3)}{\sin^{4}(x/3)} \cdot \cos^2(x/3) \, dx \]

Step 3: Use substitution to evaluate the integral in terms of u.

Substituting \( u = \sin(x/3) \) and rearranging, the integral becomes:\[ \int \frac{\cos^2(x/3)}{u^4} \, 3 \frac{du}{\cos(x/3)} = 3 \int \frac{\cos(x/3)}{u^4} \frac{\cos^2(x/3)}{\cos(x/3)} \, du = 3 \int \frac{\cos(x/3) \cdot \cos(x/3)}{u^4} \, du = 3 \int \frac{1 - u^2}{u^4} \, du \]

Step 4: Simplify the integrand and integrate with respect to u.

Split \( \frac{1-u^2}{u^4} \) into two separate fractions:\[ \frac{1}{u^4} - \frac{u^2}{u^4} = \frac{1}{u^4} - \frac{1}{u^2} \]Integrate each term separately:\[ \int \frac{1}{u^4} \, du = \int u^{-4} \, du = \frac{u^{-3}}{-3} = -\frac{1}{3u^3} \]\[ \int \frac{1}{u^2} \, du = \int u^{-2} \, du = \frac{u^{-1}}{-1} = -\frac{1}{u} \]

Step 5: Compile the results and back-substitute for x.

Combine the results from Step 4:\[ 3 \left( -\frac{1}{3u^3} + \frac{1}{u} \right) = -\frac{1}{u^3} + \frac{3}{u} \]Back-substitute \( u = \sin(x/3) \):\[ -\frac{1}{\sin^3(x/3)} + \frac{3}{\sin(x/3)} \]Thus, the evaluated integral is:\[ -\csc^3(x/3) + 3\csc(x/3) + C \]where \( C \) is the constant of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a valuable technique used to simplify integrals, especially those involving trigonometric expressions in complex forms. In this problem, trigonometric substitution plays a crucial role in converting the integrand \[ \cot^{3}(x/3) \csc(x/3) \] into a form that's easier to integrate. This is achieved by expressing the trigonometric functions solely in terms of sine and cosine:

• The cotangent is expressed as \( \cot(u) = \frac{\cos(u)}{\sin(u)} \).
• The cosecant is expressed as \( \csc(u) = \frac{1}{\sin(u)} \).

By writing everything in terms of sine and cosine, the integrand becomes: \[ \frac{\cos^{3}(x/3)}{\sin^{4}(x/3)} \]Simplifying further with the substitution \( u = \sin(x/3) \), we find \( du = \frac{1}{3} \cos(x/3) \, dx \). This allows us to express the whole integrand in terms of \( u \), dramatically simplifying the integration process.

Trigonometric Functions

Trigonometric functions are foundational in calculus and are pivotal for handling integrals involving angles. In this exercise, the primary functions used are:

• Sine (\( \sin \)): This function relates the length of the opposite side of a right triangle to its hypotenuse. It's crucial in the substitution \( u = \sin(x/3) \), a step that simplifies the integration.
• Cosine (\( \cos \)): This function provides the ratio of the adjacent side to the hypotenuse. It’s used to alter the expression into terms of sine and cosine, facilitating the integration.
• Cotangent (\( \cot \)): Defined as \( \cot(u) = \frac{\cos(u)}{\sin(u)} \), it changes into a pure fraction of sine and cosine, easing simplification.
• Cosecant (\( \csc \)): As the reciprocal of sine, \( \csc(u) = \frac{1}{\sin(u)} \), helps represent products of trigonometric functions as divisions of simple sines and cosines.

Understanding how these functions interplay allows the integration of complex expressions, making the process more accessible and systematic.

Integration Techniques

Integrating trigonometric expressions efficiently involves applying the right techniques. Here, the integral \[ \int \frac{\cos^{3}(x/3)}{\sin^{4}(x/3)} \, dx \] is tackled using known substitution methods. Key steps include:

• Transform the trigonometric expressions into simple sine and cosine terms.
• Use trigonometric substitution \( u = \sin(x/3) \) to express derivatives with simple expressions, converting a complicated integral into a manageable polynomial form.
• Split the integrand into separate integrable functions: \[ \frac{1-u^2}{u^4} \Rightarrow \frac{1}{u^4} - \frac{1}{u^2} \]
• Integrate each term individually: Techniques for integrating powers of \( u \) such as \[ \int u^{-4} \, du \text{ and } \int u^{-2} \, du \] are required, yielding antiderivatives like \[ -\frac{1}{3u^3} \text{ and } -\frac{1}{u} \].

Finally, the step of back-substituting \( u = \sin(x/3) \) gives the integral in the original variable form, including the constant of integration. These strategic steps in tackling this problem highlight how combining algebra and calculus can simplify seemingly intricate integrals.