Problem 10 - Thomas Calculus

Question

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int \csc (\pi x-1) d x $$

Step-by-Step Solution

Answer

The evaluated integral is \(-\frac{1}{\pi} \ln |\csc(\pi x - 1) + \cot(\pi x - 1)| + C.\)

Step 1: Choose a Substitution

To solve the integral \( \int \csc (\pi x - 1) \, dx \), use the substitution method. Let's set \( u = \pi x - 1 \). Then, differentiate both sides: \( du = \pi \, dx \). This implies that \( dx = \frac{du}{\pi} \). We will substitute \( u \) and \( dx \) in the integral.

Step 2: Substitute and Change of Limits

Substitute \( u = \pi x - 1 \) and \( dx = \frac{du}{\pi} \) into the integral. The integral becomes: \[ \int \csc(u) \cdot \frac{du}{\pi} = \frac{1}{\pi} \int \csc(u) \, du. \] Our integral is now easier to solve because it's a standard integral form.

Step 3: Integrate the Standard Form

We now have \( \frac{1}{\pi} \int \csc(u) \, du \). The integral of \( \csc(u) \) is \(-\ln|\csc(u) + \cot(u)| + C\), where \(C\) is the integration constant. Thus, \[ \frac{1}{\pi} (-\ln|\csc(u) + \cot(u)|) + C. \]

Step 4: Substitute Back to Original Variable

Replace \( u \) back with the original variable in terms of \( x \). Since \( u = \pi x - 1 \), our expression becomes \(-\frac{1}{\pi} \ln |\csc(\pi x - 1) + \cot(\pi x - 1)| + C. \) This is the evaluated integral for the given problem.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique often used in calculus to simplify integrals involving trigonometric functions. The key idea is to replace the original variable with a trigonometric expression. This can often make the integration process easier. In our example, the integrand is \( \csc(\pi x - 1) \). We used the substitution \( u = \pi x - 1 \). This substitution was chosen because the argument of the \( \csc \) function needed simplification for ease of integration. By setting \( u = \pi x - 1 \), we can express \( dx \) in terms of \( du \). Calculating the differential \( du = \pi dx \) leads to \( dx = \frac{du}{\pi} \).

• This substitution transforms our integral into a form that is much easier to manage.
• It often helps to reduce a complex integral into standard integrals that are easier to solve, as we'll see.

Definite Integrals

A definite integral represents the signed area under a curve, from one point to another on the x-axis. However, in our exercise, we are dealing with an indefinite integral. This means we are looking for a function, rather than a specific value. To apply substitution, we think of how the change of variables affects the limits of integration when applicable. In this specific exercise, we perform substitution to simplify the integrand. Although no specific bounds have been given (since it’s more about indefinite integration here), understanding definite integrals can be crucial when such bounds are involved:

• After substitution, you need to also change the limits from x-values to u-values.
• Remember that limits change according to the substitution equation used.
• After integration, reverting back to the original variable may affect the bounds.

This fundamental understanding helps when you do encounter definite integrals in similar problems.

Standard Integrals

Standard integrals are common, pre-derived integrals that often appear in calculations. Knowing them can save a lot of time when integrating, especially when a trigonometric substitution is involved. In our scenario, after substitution, the integral \( \int \csc(u) \, du \) directly relates to a standard integral. The antiderivative of \( \csc(u) \) is \(-\ln|\csc(u) + \cot(u)| + C\).

• Recognizing the standard integral allows you to skip steps in derivation during exams or exercises.
• Saving these forms to memory can aid immensely in solving calculus problems quickly.
• Standard integrals are often used in the final step after substitution for both trigonometric and polynomial expressions.

By shifting to the standard form via substitution, computation becomes significantly easier, as illustrated.