Question

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $$ \int \frac{x^{2} d x}{x^{2}+4} $$

Short answer

\(4x - 8\tan^{-1}\left(\frac{x}{2}\right) + C\)

Step-by-step solution

Identify the Trigonometric Substitution

We notice that the integral involves the expression \(x^2 + 4\). This resembles the form \(a^2 + x^2\), with \(a = 2\). The trigonometric substitution for this form is \(x = 2 \tan(\theta)\), which simplifies the integral.

Substitute and Simplify the Integral

Using the substitution \(x = 2 \tan(\theta)\), we find \(dx = 2 \sec^2(\theta) d\theta\). Substitute these into the integral:\[\int \frac{(2\tan(\theta))^2 \cdot 2 \sec^2(\theta) d\theta}{(2\tan(\theta))^2 + 4}\]This simplifies to:\[8 \int \frac{\tan^2(\theta) \sec^2(\theta) d\theta}{\sec^2(\theta)}\]Since \(\tan^2(\theta) \sec^2(\theta) = \sin^2(\theta) / \cos^2(\theta)\), this reduces to:\[8 \int \tan^2(\theta) d\theta\]

Solve the Integral in Terms of \(\theta\)

To simplify \(\int \tan^2(\theta) d\theta\), use the identity \(\tan^2(\theta) = \sec^2(\theta) - 1\). Thus:\[\int \tan^2(\theta) d\theta = \int (\sec^2(\theta) - 1) d\theta = \int \sec^2(\theta) d\theta - \int 1 d\theta\]The integral of \(\sec^2(\theta)\) is \(\tan(\theta)\), and the integral of \(1\) is \(\theta\), so:\[\int \tan^2(\theta) d\theta = \tan(\theta) - \theta + C\]

Express the Result in Terms of \(x\)

Recall the substitution \(x = 2 \tan(\theta)\), meaning \(\tan(\theta) = x/2\). Therefore, \(\theta = \tan^{-1}(x/2)\). Substitute back into the expression:\[8(\tan(\theta) - \theta) = 8\left(\frac{x}{2} - \tan^{-1}\left(\frac{x}{2}\right)\right)\]Thus, the integral in terms of \(x\) is:\[4x - 8\tan^{-1}\left(\frac{x}{2}\right) + C\]

Conclusion

After substituting back, we have the final result for the integral. Therefore, the integral \(\int \frac{x^2 dx}{x^2 + 4}\) evaluates to:\[4x - 8\tan^{-1}\left(\frac{x}{2}\right) + C\]

Key concepts

Integration Techniques

Integration is a fundamental concept in calculus, where it is used to find areas under curves, among other applications. Various techniques are applied depending on the form of the function. One such technique is **trigonometric substitution**, used particularly when dealing with integrals involving expressions like \(a^2 + x^2\).

Trigonometric substitution transforms complex algebraic expressions into simpler trigonometric ones. It leverages known trigonometric identities to simplify integration.

Here's how it generally works:

• Identify a pattern in the integrand, like \(a^2 + x^2\), which suggests a substitution such as \(x = a \tan(\theta)\).
• Transform the differential \(dx\) accordingly, using the derivative of the chosen substitution.
• Substitute both \(x\) and \(dx\) into the integral.
• Simplify using trigonometric identities and standard integral tables.

This technique can significantly simplify integrals that would otherwise be difficult to evaluate directly. Furthermore, by reverting to the original variable using inverse trigonometric functions, the solution is completed.

Calculus

Calculus is the mathematical study of continuous change and is divided into two main branches: *differential calculus* and *integral calculus*.
The problem at hand involves the integral calculus aspect, which focuses on accumulation and area under curves.

Integral calculus is about finding the total accumulation of values, or in other words, the integral of a function. In this exercise, we're particularly interested in indefinite integrals. These are integrals that represent a family of functions and usually include a constant \(C\) because the derivative of a constant is zero.

Using trigonometric substitution is a concrete example of how we apply integral calculus to solve non-trivial problems. By transforming the expression into a more straightforward form, the integration becomes manageable.

Moreover, calculus provides the framework, tools, and rules for maneuvering between different forms of expressions and functions to evaluate integrals accurately. By applying these principles step-by-step, even challenging integrals, like the exercise given, become less daunting.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for any value of the involved variables where both sides of the equation are defined. These identities are powerful tools in simplifying expressions during integration and solving trigonometric equations.

In the context of trigonometric substitution:

• They allow us to express \(\tan^2(\theta)\) in terms of \(\sec^2(\theta)\), which is critical to simplifying the integral.
• One key identity used is \(\tan^2(\theta) = \sec^2(\theta) - 1\). This identity helps transform complex quadratic trigonometric functions into simpler linear ones.
• Additionally, knowing that \(\sec^2(\theta)\)'s integral is \( \tan(\theta)\) directly provides a solution pathway for the integration problem.

Understanding these identities not only aids in trigonometric substitution but also simplifies many facets of calculus problems. They form the backbone of translating and breaking down complex trigonometric expressions into easier integrals to evaluate. By mastering these identities, students and practitioners can manipulate and simplify diverse expressions confidently.