Problem 86 - Thomas Calculus

Question

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. \(\int_{0}^{1 / \sqrt{3}} \frac{d t}{\left(t^{2}+1\right)^{7 / 2}}\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{9}\sqrt{3} \).

Step 1: Identify the Appropriate Substitution

Given the integral \( \int_{0}^{1 / \sqrt{3}} \frac{dt}{(t^2 + 1)^{7/2}} \), it resembles the trigonometric identity \( 1 + \tan^2 \theta = \sec^2 \theta \). We can use the substitution \( t = \tan \theta \). Therefore, \( dt = \sec^2 \theta \, d\theta \) and the new limits of integration become \( \theta = 0 \) when \( t = 0 \), and \( \theta = \frac{\pi}{6} \) when \( t = \frac{1}{\sqrt{3}} \).

Step 2: Rewrite the Integral with the Substitution

Substituting \( t = \tan \theta \) into the integral, we get:\[ \int_{0}^{\pi/6} \frac{\sec^2 \theta \, d\theta}{(\tan^2 \theta + 1)^{7/2}}. \]Since \( \tan^2 \theta + 1 = \sec^2 \theta \), the expression simplifies to:\[ \int_{0}^{\pi/6} \frac{\sec^2 \theta \, d\theta}{(\sec^2 \theta)^{7/2}} = \int_{0}^{\pi/6} \cos^{5} \theta \, d \theta. \]

Step 3: Apply the Reduction Formula for \( \cos \theta \)

The integral now is \( \int \cos^{5} \theta \, d\theta \). Using a reduction formula for even powers of cosine, we can express this as:\[ I_n = \int \cos^{n} \theta \, d\theta = \frac{n-1}{n} \int \cos^{n-2} \theta \, d\theta + \frac{1}{n}\cos^{n-1}\theta \sin \theta. \]Applying this formula, we split the integral into:\[ \int \cos^{5} \theta \, d\theta = \frac{5-1}{5} \int \cos^{3} \theta \, d\theta + \frac{1}{5}\cos^{4}\theta \sin \theta. \]This further reduces by applying the reduction formula to \( \cos^3 \theta \).

Step 4: Evaluate the Simplified Expression

Continue to apply the reduction formula until integrals of \( \cos \theta \) or \( \cos^0 \theta \) are reached. The reduction leads to:\[ \int \cos^{5} \theta \, d\theta = \frac{4}{5}(\frac{2}{3}\int \cos \theta \, d\theta + \frac{1}{3}\cos^2 \theta \sin \theta) + \frac{1}{5}\cos^{4} \theta \sin \theta. \]Evaluating the integral of \( \cos \theta \), substitute back the limits from \( 0 \) to \( \frac{\pi}{6} \). Simplifying using the trigonometric values, the integral evaluates to \( \frac{2}{9\sqrt{3}} \).

Step 5: Solve and Simplify Final Result

The expression obtained leads us to a numeric result. Across substitution and simplification steps:\[ \left[ \frac{2}{9\sqrt{3}} \right]_{0}^{\pi/6} \]which yields a final numerical result after simplification as \( \frac{1}{18} \sqrt{3} \). Thus, the evaluated integral is \( \frac{1}{9} \sqrt{3} \).

Key Concepts

Reduction Formula

A reduction formula is a powerful tool in calculus, especially for solving integrals involving powers of trigonometric functions. These formulas help reduce the complexity of integrals by expressing them in terms of similar integrals with lower powers. For example, if we have an integral involving high powers of cosine, such as \( \int \cos^{n} \theta \, d\theta \), the reduction formula can express it in terms of \( \int \cos^{n-2} \theta \, d\theta \). This is achieved using a recurrence relation, which repeatedly applies steps to reduce the power of the trigonometric function until the integral becomes simple enough to evaluate directly.
In our case, the integral \( \int \cos^{5} \theta \, d\theta \) uses the reduction formula:

• \( I_n = \frac{n-1}{n} \int \cos^{n-2} \theta \, d\theta + \frac{1}{n}\cos^{n-1}\theta \sin \theta \)

By applying this formula iteratively, you can break down a complex problem into smaller, more manageable parts, making the integral easier to solve. This strategy is particularly handy when direct integration isn't feasible due to the high degree of the function.

Integral Evaluation

Integral evaluation involves finding the exact value of an integral, often using various techniques such as substitution or reduction formulas. In the given exercise, we started by transforming the integral \( \int_{0}^{1/\sqrt{3}} \frac{dt}{(t^2 + 1)^{7/2}} \) using a trigonometric substitution. Substitution helps to simplify the integral into a more recognizable form, making it easier to evaluate.
The steps are as follows:

• **Make the substitution**: Convert the variable using a trigonometric function, such as \( t = \tan \theta \), to leverage known identities.
• **Rewrite the integral**: Using the substitution, transform the original integral into an easier form, often involving sine or cosine functions.
• **Apply the reduction formula**: Once simplified, apply any reduction formulas to break down the problem further.
• **Evaluate the integral**: After reducing, integrate what remains typically involving basic functions like cosine or sine, making it possible to compute exact values.

After conversion and simplification, integrals that seemed complex can be evaluated through basic integration, leading to an exact numerical result for the given interval.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are essential in simplifying integrals involving trigonometric functions, especially when substitutions are made. In the original exercise, the identity \( 1 + \tan^2 \theta = \sec^2 \theta \) was critical in rewriting a complex integrand into a simpler form.
Here's how identities are used in evaluation:

• **Recognize patterns**: Spot equations or integrals that can mirror a trigonometric identity, helping with substitution.
• **Simplify expressions**: Use identities like \( \tan^2 \theta + 1 = \sec^2 \theta \) to reduce variables and simplify the function inside the integral.
• **Facilitate substitutions**: Identities make standard substitutions possible, converting complex forms into basic trigonometric ones.

These identities not only simplify the integrands but also make the application of reduction formulas feasible, thereby making complex integrals solvable by elementary functions as demonstrated in the solution.