Problem 75 - Thomas Calculus

Question

Evaluate each integral in Exercises \(71-82\) by using any technique you think is appropriate. $$ \int(\csc x-\sec x)(\sin x+\cos x) d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{4} \ln |\sin 2x| + C \).

Step 1: Expand the Expression

First, we need to expand the expression \((\csc x - \sec x)(\sin x + \cos x)\). This means we distribute each term in the first parenthesis to each term in the second parenthesis.1. \(\csc x \cdot \sin x + \csc x \cdot \cos x - \sec x \cdot \sin x - \sec x \cdot \cos x\) With simplification, 2. \( (1 + \csc x \cos x - \sec x \sin x - 1) \), because \( \csc x \sin x = 1 \) and \( \sec x \cos x = 1 \). This simplifies to:3. \( \csc x \cos x - \sec x \sin x \).

Step 2: Simplify using Trigonometric Identities

We want to simplify \( \csc x \cos x - \sec x \sin x \) further if possible.1. Recall that \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \).2. Therefore, the expression becomes \( \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \).

Step 3: Combine into a Single Fraction

To integrate, it's helpful to write the expression as a single fraction.1. Find a common denominator, which in this case is \( \sin x \cos x \).2. Rewrite: \( \frac{\cos^2 x - \sin^2 x}{\sin x \cos x} \).

Step 4: Use Double Angle Identity

Use the trigonometric identity \( \cos 2x = \cos^2 x - \sin^2 x \) to simplify the numerator.1. The expression now reads \( \frac{\cos 2x}{\sin x \cos x} \).2. This can be simplified further as \( \frac{1}{2} \frac{\cos 2x}{\frac{1}{2} \sin 2x} \) using another identity, \( \sin 2x = 2 \sin x \cos x \).3. This simplifies to \( \frac{1}{2} \cot 2x \).

Step 5: Integrate the Simplified Expression

We now integrate \( \frac{1}{2} \cot 2x \).1. The integral of \( \cot 2x \) is \( \frac{1}{2} \ln |\sin 2x| + C \).2. Therefore, the integral of \( \frac{1}{2} \cot 2x \) is \( \frac{1}{4} \ln |\sin 2x| + C \).

Step 6: Conclusion

The evaluated integral is \( \frac{1}{4} \ln |\sin 2x| + C \). Make sure to verify by differentiating the result to ensure it returns to the original function.

Key Concepts

Trigonometric Identities

When tackling integrals involving trigonometric functions, trigonometric identities can be invaluable tools in simplifying expressions. In this problem, the identities for the cosecant and secant functions are particularly useful:

• The cosecant identity is \( \csc x = \frac{1}{\sin x} \).
• The secant identity is \( \sec x = \frac{1}{\cos x} \).

Using these identities allowed us to transform the terms \( \csc x \cos x \) and \( \sec x \sin x \) into simpler fractions. By rewriting trigonometric expressions using identities, we can often reveal simpler forms that make integration far more manageable.
In addition, the double angle identity \( \cos 2x = \cos^2 x - \sin^2 x \) played a critical role. By converting the two trigonometric square terms in the numerator into a double angle term, the function became easier to integrate.
These identities are basic tools in trigonometry, yet their power to simplify and solve complex problems cannot be overstated in integral calculus.

Integration by Substitution

Integration by substitution is akin to the chain rule for differentiation—it allows us to handle more complex integrals by simplifying them. This method is typically used when an integral contains a function and its derivative. However, in this particular exercise, substitution opportunities arise slightly differently.
Here, the expression was simplified down to \( \frac{1}{2} \cot 2x \) after successive rationalizations and simplifications. The transformation that resulted in \( \frac{1}{2} \cot 2x \) from \( \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \) exploits the idea of substitution to leverage a simple trigonometric function, \( \cot 2x \), that is straightforward to integrate.

• A crucial aspect of successful integration by substitution is identifying a substitution that transforms the integral into an elementary form.
• This often involves looking out for expressions that resemble derivative pairs or can be recast using simple identities.

Despite not being a textbook example of traditional substitution, recognizing and rewriting expressions through identities derived a form that withstood easy integration.

Trigonometric Integrals

Trigonometric integrals specifically deal with functions that involve products or powers of sine, cosine, and other trigonometric functions. Integrating these requires familiarity with trigonometric and sometimes hyperbolic identities.
In this exercise, after expansion and simplification, we reached a stage of integrating \( \frac{1}{2} \cot 2x \). The integral of \( \cot u \) is a recognizable form, \( \int \cot u \, du = \ln |\sin u| + C \), allowing us to render the solution straightforwardly as \( \frac{1}{4} \ln |\sin 2x| + C \).

• Key strategies include converting products into simpler terms using identities or breaking down expressions into known integrals.
• Such steps are essential when dealing with non-linear trigonometric relationships that require innovative transformations.

Mastering these techniques underpins many complex integrations and improves problem-solving versatility in calculus.