Question

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $$ \int \frac{d x}{\left(x^{2}-9\right)^{3 / 2}} $$

Short answer

The integral evaluates to \(-\frac{1}{3\sqrt{x^2 - 9}} + C\).

Step-by-step solution

Identify the Trigonometric Substitution

The integrand involves the expression \(x^2 - 9\), which suggests a substitution of the form \(x = 3\sec\theta\) because \(x^2 - a^2\) fits this trigonometric identity. Recall that \(x = a \sec \theta\) and \(a = 3\).

Differentiate the Substitution

To change variables completely, differentiate \(x = 3\sec\theta\) with respect to \(\theta\). We get \(\frac{dx}{d\theta} = 3\sec\theta\tan\theta\), or equivalently, \(dx = 3\sec\theta\tan\theta\, d\theta\).

Substitute and Simplify the Integral

Substitute \(x = 3\sec\theta\) and \(dx = 3\sec\theta\tan\theta\, d\theta\) into the integral. The integrand becomes \(\int \frac{3\sec\theta\tan\theta\, d\theta}{(9\sec^2\theta - 9)^{3/2}}\). Simplifying gives \(\int \frac{3\sec\theta\tan\theta\, d\theta}{(9\tan^2\theta)^{3/2}}\).

Simplify Further and Evaluate the Integral

Simplify the expression inside the integral: \(9(\tan^2\theta)^{3/2} = 27|\tan^3\theta|\). This simplifies the integral to \(\int \frac{3\sec\theta\tan\theta\, d\theta}{27\tan^3\theta}\), which simplifies to \(\int \frac{\sec\theta\,d\theta}{9\tan^2\theta}\). Recognize this as a solvable trigonometric integral: \(\int \frac{1}{9\sin^2\theta}\, d\theta = \int \frac{1}{9}\csc^2\theta\, d\theta\).

Integrate Using Known Trigonometric Identity

The integral \(\int \csc^2\theta\, d\theta = -\cot\theta + C\). Therefore, \(\int \frac{1}{9}\csc^2\theta\, d\theta = -\frac{1}{9}\cot\theta + C\).

Convert Back to Original Variable

Recall that \(x = 3\sec\theta\), which implies \(\tan\theta = \frac{\sqrt{x^2 - 9}}{3}\) and \(\cot\theta = \frac{3}{\sqrt{x^2 - 9}}\). Substitute back to get the expression in terms of \(x\): \(-\frac{1}{9}\cdot\frac{3}{\sqrt{x^2 - 9}} + C = -\frac{1}{3\sqrt{x^2 - 9}} + C\).

Key concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It provides tools to analyze and describe changes, both infinitesimally small and vastly large. One of its core components is derivatives, which help us understand the rate of change, similar to how speed describes a change in distance over time.
Another essential component of calculus is integrals, which are used to determine quantities like areas under curves. These calculations help us in various ways, from physics to economics, allowing us to solve real-world problems by modeling continuous change.
Trigonometric substitution is one technique used in calculus to evaluate complex integrals. It is particularly useful when dealing with expressions involving square roots, and it utilizes the known identities from trigonometry to simplify the integration process. This can make difficult problems more manageable and solvable in terms of elementary functions.

Integral Calculus

Integral calculus is concerned with the accumulation of quantities. While differential calculus deals with rates of change, integral calculus compiles those rates to find totals or areas. Think of it as putting together all the small pieces into a complete picture.
The techniques of integral calculus are invaluable for solving problems related to areas, volumes, central points, and many physical phenomena. For example, architects often use integrals to calculate the amount of materials needed for construction projects. It helps in understanding and predicting changes over continuous functions.
In integral calculus, trigonometric substitution is a method used for integrals that involve complex radical expressions. This approach converts these expressions into trigonometric identities that are easier to manage, revealing a methodical process for solving otherwise intractable integrals.

Integration Techniques

Integration techniques are methods applied to calculate the integral of functions. Since integration can sometimes be challenging, these techniques simplify the process, providing us with solutions that might otherwise be difficult to achieve.
A common technique is trigonometric substitution, where we replace variables with trigonometric functions to simplify the expression. For example, in the integral \(\int \frac{dx}{(x^2 - 9)^{3/2}}\), we recognize \(x^2 - a^2\) and use \(x = a \sec \theta\) for substitution. This step often reduces a complex algebraic integrand into a simpler trigonometric integrand, making it easier to integrate.
Other methods include integration by parts, substitution, and partial fractions. Choosing the appropriate technique depends on the structure of the function we are integrating. Mastering these strategies is essential for students of calculus, giving them a toolbox to approach and solve a wide range of problems.