Problem 24 - Essential Calculus

Question

Find the integral involving secant and tangent. $$ \int \tan ^{3} \frac{\pi x}{2} \sec ^{2} \frac{\pi x}{2} d x $$

Step-by-Step Solution

Answer

The integral of \(\tan ^{3} \frac{\pi x}{2} \sec ^{2} \frac{\pi x}{2} dx\) is \(\frac{1}{4\pi}\tan^{2}(\frac{\pi x}{2}) + C\).

Step 1: Substitute the variable

Set \(u = \tan(\frac{\pi x}{2})\). Hence, \(\frac{du}{dx} = \sec^2(\frac{\pi x}{2}) * \frac{\pi}{2}\) and \(dx = \frac{2 du}{\pi \sec^2(\frac{\pi x}{2})}\). Replace these in the integral, we will get \(\int u^{3} du = \frac{1}{4}\int u^{3} \frac{2du}{\pi u^{2}}\).

Step 2: Cancel common terms

In the above expression, cancel the common terms, which gives us \(\frac{1}{2\pi}\int u du\).

Step 3: Integrate and substitute back

Now, perform the integration: \(\frac{1}{2\pi} * \frac{1}{2}u^{2} = \frac{1}{4\pi}u^{2}\). Substituting back, we get \(\frac{1}{4\pi}\tan^{2}(\frac{\pi x}{2})\).

Step 4: Write the final answer

The integral is solved and the final answer is: \(\frac{1}{4\pi}\tan^{2}(\frac{\pi x}{2}) + C\), where \(C\) is the constant of integration.

Key Concepts

Integration Techniques

In the world of integral calculus, integration techniques are a set of tools that help us find antiderivatives or calculate integrals of complex functions. One common technique is substitution, where we simplify the integral by changing variables. This transforms the problem into something more manageable.
For the integral \( \int \tan^{3} \frac{\pi x}{2} \sec^{2} \frac{\pi x}{2} \ dx \), substitution is particularly helpful because of the trigonometric functions involved. By choosing an appropriate substitution, we can often reduce a complicated integral into a simpler form. 

In this example, substitution allows us to focus on integrating a polynomial function \(u^3\), which is more straightforward. Other techniques include integration by parts, partial fractions, and trigonometric identities, but substitution is well-suited for problems involving specific functions like secant and tangent.

Trigonometric Functions

Trigonometric functions like tangent and secant are often involved in calculus problems. These functions can make integration more challenging, but they also offer opportunities for simplification through substitutions or identities.
The function \(\tan(x)\) is the ratio of \(\sin(x)\) to \(\cos(x)\), and \(\sec(x)\) is the reciprocal of \(\cos(x)\). These relationships are useful when manipulating integrals, as they allow us to convert between various trigonometric forms.

Recognizing these functions' derivatives is key to solving integrals that include them. The derivative of \(\tan(x)\) is \(\sec^2(x)\), which is directly utilized in the substitution step of our solution. By understanding how these functions relate to one another and their derivatives, we can integrate effectively even when faced with complex expressions.

U-Substitution

U-substitution is a technique used in integration, particularly effective when dealing with composite functions. The principle is simple: identify a part of the integrand that can be replaced with a single variable, usually \(u\), making the integral easier.
In the example \(\int \tan^{3} \frac{\pi x}{2} \sec^{2} \frac{\pi x}{2} \ dx \), we set \(u = \tan(\frac{\pi x}{2})\). This choice simplifies the integral because the differential \(du\) is linked directly to \(\sec^2\), the derivative of \(\tan(x)\).

The substitution transforms our integrand into a power function, \(u^3\), which is easier to integrate: \(\frac{1}{2\pi}\int u \, du\). Finally, we substitute back the original variable and its expression in terms of \(x\) to finalize the integral. U-substitution reduces the workload and streamlines the process of solving integrals with non-linear functions.