Question

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \csc x d x\)

Short answer

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

Step-by-step solution

Rewrite the Integral

Note that the integral involves the cosecant function, which can be rewritten using its definition. We have \( \csc x = \frac{1}{\sin x} \). Thus, the integral becomes: \[ \int \csc x \, dx = \int \frac{1}{\sin x} \, dx. \]

Use Trigonometric Identity

Recall that \(\csc x\) can be expressed in terms of a fraction of sine and cosine functions: \[ \csc x = \frac{1}{\sin x} = \frac{\sin x}{\sin^2 x}, \] which can be beneficial in recognizing the derivatives involved.

Simplify Further

The integral \( \int \frac{1}{\sin x} \, dx \) is typically approached by using an alternative method that involves multiplying by a cleverly chosen form of 1. Multiply by \( \frac{\csc x + \cot x}{\csc x + \cot x} \): \[ \int \csc x \, dx = \int \frac{1}{\sin x} \, \cdot \frac{\csc x + \cot x}{\csc x + \cot x} \, dx. \]

Apply U-substitution

Let \( u = \csc x + \cot x \). Then, the derivative is \( du/dx = -\csc x \cot x - \csc^2 x \), which implies that \[ du = (-\csc x \cot x - \csc^2 x) \, dx. \]Notice that this is similar to our integrand, and can be simplified:\[ dx = -\frac{du}{\csc x + \cot x}. \]

Integrate with Substitution

Now integrate with respect to \( u \):\[-\int \frac{du}{u}.\]The result of this integration is:\[ -\ln |u| + C. \]

Substitute Back

Replace \( u \) back with \(\csc x + \cot x\) to complete the process:\[ -\ln |\csc x + \cot x| + C. \]

Verify Solution with CAS

Use a Computer Algebra System (CAS) to check the integration result. Input the original integral \( \int \csc x \, dx \) and confirm that the CAS outputs: \[ -\ln |\csc x + \cot x| + C. \] This verifies the manual solution is correct.

Key concepts

Cosecant Integration

Integrating the cosecant function can be a bit tricky, especially if you're just getting familiar with trigonometric integrals. The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \). To integrate \( \csc x \), we start by rewriting it in a form that's easier to manage: \( \int \csc x \ dx = \int \frac{1}{\sin x} \ dx \). This simplification allows us to explore further techniques, like multiplying by a clever choice of 1, which can transform the integral into a format that's more straightforward to solve.

Trigonometric Identities

Trigonometric identities are a crucial component in solving integrals involving trigonometric functions. They help simplify expressions and reveal hidden patterns in integrals. One powerful strategy is to express trigonometric functions in terms of each other. For instance, recognizing that \( \csc x = \frac{1}{\sin x} \) can sometimes be simplified further using a combination of sine and cosine identities. These identities are not only handy for integration, but they're also the backbone of many calculations involving angles and waves. Knowing them well allows you to manipulate and simplify trigonometric expressions, making the integration process much easier. In this problem, the identity aids us in molding the integral into a solvable form.

U-substitution

The technique of \( u \)-substitution is used extensively in calculus to simplify the process of integration. It's particularly useful for integrals that involve compositions of functions. The idea is to substitute part of the integrand with a single variable \( u \), essentially turning a complex integral into a simpler one that is easier to evaluate. In our case, letting \( u = \csc x + \cot x \) helps in transforming the integral into a more manageable form. This transformation works because the derivative of \( u \) is closely related to the original integrand. This allows the integral to be solved as \(-\int \frac{du}{u} \), which simplifies to \(-\ln |u| + C\). Once integration is complete, we substitute back \( u = \csc x + \cot x \) to express the solution in terms of \( x \). This technique is vital for many integration problems, as it converts a challenging integral into a form that is straightforward to handle.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a powerful tool that assists in solving, verifying, and visualizing mathematical equations and integrals. These systems utilize algorithms to quickly find solutions to complex problems involving calculus, algebra, and more. When you reach a solution manually, like \(-\ln |\csc x + \cot x| + C\) in this case, a CAS can be employed to verify its correctness.Using a CAS, you enter the original integral, and the system outputs a solution, helping confirm the steps taken and the accuracy of the solution. While learning, CAS provides a strong assurance for students, showing that their manual work aligns with computational results. Moreover, it offers an opportunity to explore further variations and scenarios of a given problem efficiently, fostering deeper understanding and learning.