Problem 51 - Calculus Early Transcendentals

Question

Use a change of variables to evaluate the following definite integrals. $$\int_{1 / 3}^{1 / \sqrt{3}} \frac{4}{9 x^{2}+1} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{1 / 3}^{1 / \sqrt{3}} \frac{4}{9 x^{2}+1} d x$$ using a suitable substitution. Answer: The exact value of the definite integral is $$\frac{4}{3}[\arctan(\frac{\sqrt{3}}{3})-\arctan(\frac{1}{3})].$$

Step 1: Choose a suitable substitution

Let's choose a substitution that will simplify the denominator in the given integral. Here, we set \(u = 3x\), so \(x = \frac{u}{3}\). Now, differentiate both sides with respect to \(x\) to find the corresponding \(dx.\) $$ \frac{d u}{d x} = 3.$$ Now, solve for \(dx\): $$ dx = \frac{du}{3}.$$

Step 2: Rewrite the integral in terms of the new variable

Substitute the \(x\) and \(dx\) in the integral with the expressions in terms of \(u,\) $$\int_{1 / 3}^{1 / \sqrt{3}} \frac{4}{9 x^{2}+1} d x = \int \frac{4}{9(\frac{u}{3})^{2}+1} \cdot \frac{du}{3} = \int \frac{4}{u^2+9} du$$ Calculate the new limits of integration. At lower limit, \(x = \frac{1}{3}\): $$u = 3x = 3 \cdot \frac13 = 1$$ At upper limit, \(x = \frac{1}{\sqrt{3}}\): $$u = 3x =3 \cdot \frac{1}{\sqrt{3}} = \sqrt{3}$$ So, we have: $$\int_{1/3}^{1/\sqrt{3}} \frac{4}{9x^2+1}dx=\int_1^{\sqrt{3}}\frac{4}{u^2+9}du$$

Step 3: Integrate the function in terms of new variable

We recognize that the integral is now in the form of the arctangent function. To see that, we can recall the derivative of arctangent: $$\frac{d}{du}\arctan(\frac{u}{3}) = \frac{1}{1+(\frac{u}{3})^2} \cdot \frac{3}{3} = \frac{3}{u^2+9}$$ Then, the integral we need to solve is: $$\int \frac{4}{u^2+9}du = \frac{4}{3}\int \frac{3}{u^2+9}du = \frac{4}{3}\arctan(\frac{u}{3}) + C$$

Step 4: Apply the limits of integration to find the definite integral

Now, we just need to evaluate the antiderivative at the new limit points and subtract them. $$\int_{1/3}^{1/\sqrt{3}} \frac{4}{9x^2+1}dx=\int_1^{\sqrt{3}}\frac{4}{u^2+9}du = \frac{4}{3}[\arctan(\frac{\sqrt{3}}{3})-\arctan(\frac{1}{3})]$$ That is the exact value of the definite integral.

Key Concepts

Change of Variables

The change of variables technique, also known as substitution, is a powerful method used in calculus to simplify definite integrals. It's particularly useful when the direct evaluation of an integral is complex or cumbersome.

The basic idea involves replacing a variable in the integral with another variable that simplifies the expression. Here's how it works in steps:

• **Choose a Substitution**: Pick a new variable, often denoted as \(u\), to replace part of the original variable. This choice depends on the form of the integral. In the given exercise, setting \(u = 3x\) simplifies the denominator.
• **Differentiate**: Find the derivative \(du/dx\) and solve for \(dx\). For \(u = 3x\), we find \(dx = du/3\).
• **Rewrite the Limits**: Substitute the new variable into the limits of integration as well. You must recalculate these limits in terms of \(u\). For example, \(u = 1\) when \(x = 1/3\), and \(u = \sqrt{3}\) for \(x = 1/\sqrt{3}\).
• **Transform the Integral**: Substitute all \(x\) terms and the limits into the integral to express it fully in terms of \(u\).

By using substitution, complex integrals become simpler and more manageable, paving the way for step-by-step solutions.

Definite Integrals

A definite integral is an integral with specific limits, providing a numerical result that represents the accumulated area under a curve between two points on the axis. Definite integrals are a fundamental concept in calculus, often representing physical quantities like distance, area, or total accumulation of a rate over time.

To compute a definite integral:

• **Set Up the Integral**: Begin by determining the function to integrate and the limits. In the exercise, the function is \( \frac{4}{9x^{2}+1} \), and the limits are \( 1/3 \) to \( 1/\sqrt{3} \).
• **Apply Substitution**: Use substitution to simplify the integral, as discussed earlier. This step can sometimes transform a complex integral into a standard form that's easier to evaluate.
• **Integrate the Function**: Compute the antiderivative of the transformed integrand. Using known integration formulas, we find that it aligns with a form solvable by arctangent in this case.
• **Evaluate the Result**: Substitute the upper and lower limits into the antiderivative. Subtract the lower limit result from the upper limit to obtain the integral's value.

Note that unlike indefinite integrals, which represent a family of functions, definite integrals provide a singular numerical result.

Trigonometric Substitution

Trigonometric substitution is a technique that uses trigonometric identities to simplify the integration of expressions involving square roots and quadratics. It is especially advantageous when the integral comprises expressions such as \(a^2 + x^2\), \(a^2 - x^2\), or \(x^2 - a^2\).

This method involves:

• **Choosing a Trigonometric Function**: Select a substitution that matches the expression's form. For instance, substitute \(x = a \tan\theta\) when dealing with expressions like \(a^2 + x^2\).
• **Transform the Integral**: Change all instances of \(x\) to the trigonometric expression \(a \tan\theta\) or equivalent and adjust \(dx\) accordingly. This transforms the integral into an expression involving trigonometric functions.
• **Evaluate Trigonometric Integral**: Integrate using trigonometric identities, and eventually apply the inverse trigonometric function. The integral in the original step is simplified using the identity for arctangent derivatives, which fits this category.
• **Back Substitution**: Convert the results back to the original variable, if necessary, using the inverse trigonometric relationships.

This approach systematically reduces the problem to a simpler form and highlights the utility of trigonometric identities in solving integrals.