Problem 46 - Calculus: Graphical, Numerical, Algebraic

Question

\(\int \csc x d x \quad(\)Hint\(:\) Multiply the integrand by \(\frac{\csc x+\cot x}{\csc x+\cot x}\) and then use a substitution to integrate the result.)

Step-by-Step Solution

Answer

\(-\ln|\csc(x) + \cot(x)|\).

Step 1: Multiply the integrand by trick fraction

Multiply the integrand by \(\frac{\csc(x) + \cot(x)}{\csc(x) + \cot(x)}\), which is essentially multiplying by 1, so it doesn't change the value of the rational function. However, it allows us to rewrite the function in a more convenient way. After this multiplication, the integral becomes \(\int \frac{\csc^2(x) + \csc(x)\cot(x)}{\csc(x) + \cot(x)} d x\).

Step 2: Use substitution

Let's define a new variable \(u = \csc(x) + \cot(x)\). Then, the derivative \(du\) can be written as \(-\csc(x)\cot(x) - \csc^2(x) dx\), if we look at the numerator of the resulting integral from step 1, we can see that this is the same as the negative of our \(du\). Thus, the integral can be written in terms of \(u\) as \(-\int \frac{du}{u}\).

Step 3: Apply logarithmic integration

The integral \(-\int \frac{du}{u}\) is a simple integral that corresponds to the log function. Thus, \(-\int \frac{1}{u} du = -\ln|u|\).

Step 4: Substitute back original function

Finally, replace \(u\) with \(\csc(x) + \cot(x)\) in the solution to get the final answer. This yields the result \(-\ln|\csc(x) + \cot(x)|\).

Key Concepts

Trigonometric Integrals

Trigonometric integrals are a key component in integral calculus as they involve integrating expressions with trigonometric functions. In this exercise, we encounter the task of integrating the cosecant function, \( \csc x \). Cosecant is the reciprocal of the sine function, and direct integration of such functions is not always straightforward.The trick to simplifying these integrals often involves using trigonometric identities or clever algebraic manipulations.

• For \( \int \csc x\, dx \), we employ a tactic involving multiplying by the conjugate, \( \frac{\csc x + \cot x}{\csc x + \cot x} \).
• This technique allows us to rewrite the function in a way that is easier to integrate.

This approach is powerful because it utilizes the properties of trigonometric functions to transform the problem into a more familiar form, allowing us to use methods such as substitution more effectively.

Substitution Method

The substitution method is an essential technique in calculus that simplifies integrals by substituting a new variable for a more complex expression. In this exercise, we make use of a substitution to facilitate the integration of the trigonometric function.After rewriting the integral involving \( \csc x \) by multiplying by the conjugate, we define a new variable:

• Let \( u = \csc x + \cot x \).

This substitution is chosen carefully because its derivative, \( du = -\csc(x)\cot(x) - \csc^2(x) \, dx \), matches the numerator obtained after multiplying by the conjugate.Once substituted, the integral transforms into the simpler form:\[ -\int \frac{du}{u} \] This illustrates the utility of the substitution method, as it reduces the complexity of the integral, making it possible to apply standard integration techniques.

Logarithmic Integration

Logarithmic integration refers to the integration of fractions of the form \( \int \frac{1}{u} du \), which result in a logarithmic function. In the context of this exercise, once we have substituted the variable, we encounter such a fraction.After substitution, our integral simplifies to:\[ -\int \frac{du}{u} \] This is a straightforward case of logarithmic integration, resulting in the logarithm of the variable \( u \):\[ -\ln|u| \]Finally, we substitute back the expression for \( u \) in terms of \( x \) to complete the integration process:\[ -\ln|\csc x + \cot x| \]Logarithmic integration is powerful because of its ability to handle rational functions in calculus, providing elegant solutions to otherwise complex integrals.