Question

A secant reduction formula Prove that for positive integers \(n \neq 1\) $$ \int \sec ^{n} x d x=\frac{\sec ^{n-2} x \tan x}{n-1}+\frac{n-2}{n-1} \int \sec ^{n-2} x d x $$ (Hint: Integrate by parts with \(u=\sec ^{n-2} x\) and \(d v=\sec ^{2} x d x\) )

Short answer

Question: Prove the secant reduction formula given by $$ \int \sec^nx dx =\frac{\sec^{n-2}x\tan{x}}{n-1}+\frac{n-2}{n-1} \int \sec^{n-1}x dx $$ for positive integers \(n \neq 1\). Answer: To prove the secant reduction formula, we can follow these steps: 1. Apply integration by parts with \(u = \sec^{n-2}x\) and \(dv = \sec^2x dx\), and compute \(du\) and \(v\). 2. Simplify the result obtained after applying the integration by parts formula. 3. Isolate the integral of \(\sec^nx dx\) and obtain the final reduction formula as given. Following these steps, we can prove that $$ \int \sec^nx dx =\frac{\sec^{n-2}x\tan{x}}{n-1}+\frac{n-2}{n-1} \int \sec^{n-1}x dx $$ for positive integers \(n \neq 1\).

Step-by-step solution

Apply Integration by Parts

To prove the given formula, we will start by applying integration by parts to the integral of \(\sec^nx dx\). Since the exercise provides the hint to choose \(u = \sec^{n-2}x\) and \(dv = \sec^2{x} dx\), we need to find \(du\) and \(v\). Differentiating \(u\) with respect to \(x\), we get $$ du = (n-2)\sec^{n-3}x\sec{x}\tan{x} dx $$ Integrating \(dv\) with respect to \(x\), we get $$ v = \int \sec^2{x} dx = \tan{x}$$ Now, we use the integration by parts formula: $$ \int u dv = uv - \int v du$$ Substitute our chosen values of \(u\), \(v\), \(du\), and \(dv\):

Simplify the Result

Now, we have $$ \int \sec^nx dx = \sec^{n-2}x\tan{x} - (n-2)\int \sec^{n-3}x\sec{x}\tan^2x dx $$ We know that \(\tan^2{x} = \sec^2{x} - 1\). Replacing, we have $$ \int \sec^nx dx = \sec^{n-2}x\tan{x} - (n-2)\int \sec^{n-1}x(\sec^2{x} - 1) dx $$ Distribute the integral to get two different integrals: $$ \int \sec^nx dx = \sec^{n-2}x\tan{x} - (n-2)\int \sec^{n+1}x dx + (n-2)\int \sec^{n-1}x dx $$ Now, we want to isolate the original integral \(\int \sec^nx dx\), so we move the term with the integral of \(\sec^{n+1}x dx\) to the other side of the equation:

Isolate the Integral and Finish

After isolating the integral of \(\sec^nx dx\), we obtain $$ \int \sec^nx dx + (n-2)\int \sec^{n+1}x dx = \sec^{n-2}x\tan{x} + (n-2)\int \sec^{n-1}x dx $$ Finally, divide both sides of the equation by \((n-1)\): $$ \int \sec^nx dx =\frac{\sec^{n-2}x\tan{x}}{n-1}+\frac{n-2}{n-1} \int \sec^{n-1}x dx $$ Now, we have successfully proved the secant reduction formula for positive integers \(n \neq 1\).

Key concepts

Integration by Parts

Integration by parts is a powerful technique in calculus used to integrate products of two functions. This method is based on the product rule for differentiation and is useful when integrating the product of algebraic and trigonometric functions, among others. The formula for integration by parts is expressed as \[ \int u dv = uv - \int v du \]where \( u \) and \( dv \) are functions of a variable, usually \( x \).

To apply this technique successfully, one must correctly choose what to set as \( u \) and \( dv \), since the choice can greatly simplify the problem. In the secant reduction formula problem, the hint guides the choice of \( u = \sec^{n-2}x \) and \( dv = \sec^2{x} dx \), deliberately aimed at simplifying the integral. After making these choices, we differentiate \( u \) to find \( du \), and integrate \( dv \) to find \( v \). The original integral is then broken down into parts that are generally more manageable to integrate. This technique is particularly effective when reducing the power of trigonometric integrals, making it an essential tool for solving complex calculus problems.

Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions such as sine, cosine, secant, and tangent. These integrals often appear in calculus and can present challenges due to the nature of these functions. To integrate trigonometric functions effectively, one must be familiar with trigonometric identities, like\[ \tan^2{x} = \sec^2{x} - 1 \]which can be used to simplify integrands and enable further integration steps.

In the secant reduction formula, after applying integration by parts, we encounter a new integral with a \( \tan^2{x} \) term. Utilizing the trigonometric identity above, we transform \( \tan^2{x} \) into \( \sec^2{x} - 1 \), thus facilitating the separation of the integral into more tractable parts. Mastering such identities is crucial, as it can often mean the difference between an integral that is solvable and one that seems insurmountable. Moreover, the process of repeatedly applying these identities often leads to a pattern, as seen in the reduction formula, which ultimately aids in finding a general solution for a class of integrals.

Calculus Techniques of Integration

Various techniques of integration are applied in calculus to find the antiderivative or the integral of functions. These techniques include, but are not limited to, substitution, integration by parts, partial fractions decomposition, and specific methods for trigonometric, exponential, and logarithmic integrals. Understanding when and how to use these methods effectively is essential for tackling a wide range of integrals.

In the context of proving the secant reduction formula, integration by parts is utilized initially to break down the integral. Following that, trigonometric identities are used to simplify the expression further. The strategy here involves not just knowing the techniques, but also applying them iteratively and in the correct order to simplify the integrand. Moreover, recognizing patterns as they emerge during the integration process is a skill that enables mathematicians to create general formulas which can solve whole families of integrals, as is the case with the proved reduction formula. This skillset is fundamental in calculus and continues to be a focal point in the study and application of higher mathematics.