Problem 60 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{d x}{(x+3) \sqrt{(x+3)^{2}-25}}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{5} \sec^{-1}\left(\frac{x+3}{5}\right) + C \).

Step 1: Identify the Form of the Integral

The integral resembles the form \( \int \frac{d u}{u\sqrt{u^2-a^2}} \), which suggests that we are dealing with a trigonometric substitution integral. We need to look for a substitution that simplifies \( \sqrt{(x+3)^2 - 25} \).

Step 2: Make an Appropriate Substitution

Let \( u = x + 3 \). Then, \( du = dx \) and the expression under the square root becomes \( \sqrt{u^2 - 25} \). This suggests using the trigonometric identity \( \sec^2\theta - 1 = \tan^2\theta \). Thus, substitute \( u = 5\sec\theta \).

Step 3: Calculate the Derivatives and Substitutions

With \( u = 5\sec\theta \), we have \( du = 5\sec\theta\tan\theta\, d\theta \). The integral becomes \( \int \frac{5\sec\theta\tan\theta\, d\theta}{5\sec\theta \cdot 5\tan\theta} \), which simplifies to \( \int \frac{d\theta}{5} \).

Step 4: Solve the Simplified Integral

The integral \( \int \frac{1}{5} d\theta \) is straightforward. It evaluates to \( \frac{1}{5}\theta + C \), where \( C \) is the integration constant.

Step 5: Reverse the Substitution

To reverse the substitution, remember \( \cos\theta = \frac{5}{u} \). Thus, \( \theta = \sec^{-1}\left(\frac{u}{5}\right) = \sec^{-1}\left(\frac{x+3}{5}\right) \). Therefore, the integrated result in terms of \( x \) is \( \frac{1}{5} \sec^{-1}\left(\frac{x+3}{5}\right) + C \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify the integration process. It is particularly useful when dealing with integrals containing square roots of quadratic expressions. The idea is to substitute a trigonometric function for a variable to exploit certain trigonometric identities, which can simplify the integral.

In the exercise given, the substitution revolves around solving the square root expression \( \sqrt{(x+3)^2 - 25} \). This expression matches the pattern of \( \sqrt{u^2 - a^2} \), which suggests using the identity \( \sec^2\theta - 1 = \tan^2\theta \). By choosing \( u = 5\sec\theta \), the expression \( \sqrt{u^2 - 25} \) transforms into \( 5\tan\theta \), which greatly simplifies the integration.

When you substitute back after integration, ensure to reverse the trigonometric identities and original substitution to express the answer in terms of the variable \( x \). This requires knowledge of inverse trigonometric functions, such as \( \sec^{-1} \), which completes the extraction from the trigonometric identity.

Definite Integrals

Although the exercise provided focuses on an indefinite integral, understanding definite integrals is crucial. Definite integrals calculate the area under a curve between two points. They are presented in the form \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. This contrasts with indefinite integrals, which require finding the antiderivative without limits.

The solution to a definite integral provides a numerical value representing the area. The fundamental theorem of calculus connects differentiation and integration by stating that if \( F(x) \) is an antiderivative of \( f(x) \), then:

• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

This shows the relationship between finding an antiderivative and applying the integration limits to find the area under \( f(x) \). When working with trigonometric substitutions in definite integrals, it involves applying the substitutive integration technique and then evaluating the antiderivative within the given limits.

Antiderivatives

Antiderivatives are functions that reverse the process of differentiation. If \( F(x) \) is an antiderivative of \( f(x) \), then the derivative of \( F(x) \) is \( f(x) \). In integral calculus, finding the antiderivative is essential for solving indefinite integrals, where expressions are evaluated without specific boundaries.

For the given homework problem, the primary goal is to find the antiderivative of the integral \( \int \frac{dx}{(x+3)\sqrt{(x+3)^2-25}} \). The process involves breaking down the trigonometric substitution, substituting back to the original terms, and expressing the result along with the constant \( C \). The solution's antiderivative is \( \frac{1}{5} \sec^{-1}\left(\frac{x+3}{5}\right) + C \), showcasing how to reverse engineer to express it in terms of \( x \).

Understanding antiderivatives also leads to mastering the use of constants. The constant \( C \) in indefinite integrals reflects a family of functions, all of which have the same derivative. This constant disappears in definite integrals as specific numeric values replace it upon applying integration limits.