Question

Find the integral. Use a computer algebra system to confirm your result. $$ \int \cot ^{3} 2 x d x $$

Short answer

The solution for the integral of \(\cot^{3} 2x\) is \(-\frac{1}{2}\csc 2x + \frac{1}{2}\ln |\sin 2x|\)

Step-by-step solution

Trigonometric Identity

It is known that \(\cot^{2} x = \csc^{2} x - 1\). Therefore substitute \(\cot^{2} 2x \) with \(\csc^{2} 2x - 1\) in the integral, so the integral \(\int \cot^{3} 2x dx\) can be written as \(\int \cot 2x(\csc^{2} 2x - 1) dx\).

Separate into Two Integrals

The integral can be broken down into two integrals:\(\int \cot 2x \csc^{2} 2x dx\) & \(\int \cot 2x dx\). This gives us easier terms to work with.

Integrate

The integral of \(\int \cot 2x \csc^{2} 2x dx\) is \(-\frac{1}{2}\csc 2x\). The integral of \(\int \cot 2x dx\) is \(\frac{1}{2}\ln |\sin 2x|\).

Combine Steps

Putting steps 3 together we get: \(-\frac{1}{2}\csc 2x + \frac{1}{2}\ln |\sin 2x|\). This is our solution.

Key concepts

Trigonometric Identity

Understanding trigonometric identities can greatly simplify the process of solving complex trigonometric integrals. One fundamental identity is \( \cot^{2}x = \csc^{2}x - 1 \). By recognizing that cotangent can be expressed in terms of cosecant, we can decompose the original problem into simpler parts.

For example, if an integral involves \( \cot^{3}x \), as in our exercise, this identity allows us to rewrite the integrand as \( \cot x(\csc^{2}x - 1) \), thus converting a higher power trigonometric function into products of functions that are easier to integrate. It's essential to have a good grasp of these identities, as they can be the key to solving many trigonometric integrals effectively.

Integral of Cotangent

The integral of cotangent \( \int \cot xdx \) is a standard integral in trigonometry with a well-known solution: \( \ln|\sin x| \). But when dealing with higher powers of cotangent, such as \( \cot^3x \), the integral becomes more complex.

In our exercise, we use the identity stated earlier to express \( \cot^3x \) in a form that aids in integration. Breaking it down into simpler integrals is a useful approach: one involving \( \cot x \csc^{2}x \) and another with just \( \cot x \). Understanding how to integrate these basic functions separately allows us to later combine these partial results to find the solution to the original problem.

Computer Algebra System

A computer algebra system (CAS) is a software tool that is capable of performing symbolic mathematics. Using a CAS to confirm solutions to complex integrals can be very helpful to validate the manual calculations. After arriving at a potential solution through step-by-step integration, inputting the integral into a CAS should yield the same result if done correctly.

For students, this is an invaluable resource for checking their work. While some might argue it can be used as a crutch, when used responsibly, a CAS can actually aid in learning by confirming the correct application of integration techniques, and can provide insight into alternative methods of solving the integral.

Integration Techniques

Several integration techniques are essential tools in a mathematician's toolkit. Breaking down complex integrands into simpler ones, as done in our exercise, is a common strategy known as 'method of decomposition'.

Other techniques include integration by parts, substitution, and partial fractions. Each technique has its own applications and is used based on the type of function or integral one is faced with. Mastery of these techniques allows for the solving of a wide variety of integrals beyond basic formulas, ultimately enabling students to tackle more challenging problems with confidence.