Problem 51 - Calculus Early Transcendentals

Question

Evaluate the following definite $$\int_{0}^{1 / \sqrt{3}} \sqrt{x^{2}+1} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{0}^{1 / \sqrt{3}} \sqrt{x^{2}+1} d x$$. Answer: $$\frac{8}{3}$$

Step 1: Identify the substitution method

We can make a substitution for solving the integral. We notice that the function inside the square root is in the form of $$x^2 + 1$$. To simplify this expression, we use the trigonometric substitution $$x = \tan(\theta),$$ which also leads to $$dx = \sec^2(\theta) d\theta$$.

Step 2: Make the substitution

Now we substitute $$x$$ and $$dx$$ in the integral with the expressions above: $$\int_{0}^{1 / \sqrt{3}} \sqrt{x^{2}+1} d x = \int_{0}^{\pi/3} \sqrt{\tan^2(\theta)+1} \cdot \sec^2(\theta) d\theta$$

Step 3: Simplify the integral

Using the trigonometric identity $$\tan^2(\theta) + 1 = \sec^2(\theta)$$, we can simplify the integral: $$\int_{0}^{\pi/3} \sqrt{\tan^2(\theta)+1} \cdot \sec^2(\theta) d\theta = \int_{0}^{\pi/3} \sec^2(\theta) \cdot \sec^2(\theta) d\theta = \int_{0}^{\pi/3} \sec^4(\theta) d\theta$$

Step 4: Finding the antiderivative for the new function

Using the integration formula for the power of secant functions, we get: $$\int \sec^4(\theta) d\theta = \frac{1}{3}\sec^3(\theta)\tan(\theta) + \frac{1}{3}\tan(\theta) + C$$

Step 5: Replacing the limits of the integral

Now, we apply the Fundamental Theorem of Calculus to evaluate the definite integral: $$\int_{0}^{\pi/3} \sec^4(\theta) d\theta = \left[\frac{1}{3}\sec^3(\theta)\tan(\theta) + \frac{1}{3}\tan(\theta)\right]_0^{\pi/3}$$

Step 6: Evaluating the definite integral

Now, we substitute the upper and lower limits in the integral expression: $$\frac{1}{3}\sec^3(\pi/3)\tan(\pi/3) + \frac{1}{3}\tan(\pi/3) - \left[\frac{1}{3}\sec^3(0)\tan(0) + \frac{1}{3}\tan(0)\right] = \frac{1}{3}\sqrt{3}^3\cdot\sqrt{3} + \frac{1}{3}\sqrt{3} - 0$$ Simplifying the expression, we get the final answer: $$\int_{0}^{1 / \sqrt{3}} \sqrt{x^{2}+1} d x = \frac{8}{3}$$

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a useful technique for solving integrals that involve square roots of expressions like \(x^2 + a^2\), \(x^2 - a^2\), or \(a^2 - x^2\). In our example, we have an integral with \(\sqrt{x^2 + 1}\). A common strategy is using trigonometric functions because they can simplify these types of expressions.

When you see \(x^2 + 1\), think about the trigonometric identity \(\tan^2(\theta) + 1 = \sec^2(\theta)\). By letting \(x = \tan(\theta)\), you can transform \(x^2 + 1\) into a more manageable form. With this substitution, \(x^2 + 1 = \sec^2(\theta)\) and \(dx = \sec^2(\theta)d\theta\). This form allows us to simplify our integral significantly because dealing with \(\sec^2(\theta)\) or higher powers of it is often more straightforward.

• Choose substitution based on the structure of the integrand.
• Use trigonometric identities to simplify the expression post-substitution.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differentiation and integration, two core concepts in calculus. It essentially states that if you can find the antiderivative of a function within an integral, you can evaluate a definite integral using that antiderivative.

In this exercise, after transforming and simplifying the integral to \(\int \sec^4(\theta) d\theta\), the next step is to find its antiderivative. Once that is accomplished, the Fundamental Theorem allows us to evaluate the definite integral by plugging in the upper and lower limits into the antiderivative. This process turns a potentially difficult numerical calculation into a straightforward substitution problem.

• First, find the antiderivative of the integrand.
• Apply the upper limit and subtract the value at the lower limit.

This theorem is powerful because it simplifies the evaluation of definite integrals significantly, especially when combined with substitution techniques.

Integration Techniques

Integration techniques are methods or strategies used to solve integrals, especially when they are not straightforward. For definite integrals involving more complex forms, different techniques can be applied.

In this example, the use of trigonometric substitution and then finding the antiderivative of \(\sec^4(\theta)\) were key techniques in our solution. Identifying that \(\sec^4(\theta)\) can be separated into more easily integrable parts or using known integration formulas for powers of secant functions is critical. Another common method in these situations might include integration by parts or partial fraction decomposition, but they are not needed in this specific context.

• If a direct antiderivative is difficult to find, consider breaking down the integrand.
• Use known formulas or identities to simplify the integration process.

Understanding when and how to apply these various integration strategies makes solving complex integrals more manageable.