Question

Perform the indicated integrations. $$ \int \frac{x}{x^{4}+4} d x $$

Short answer

\( \arctan\left(\frac{x^2}{2}\right) + C \)

Step-by-step solution

Identify the integration technique

The integral \( \int \frac{x}{x^{4}+4} \, dx \) suggests a substitution might simplify it, especially because the numerator is the derivative of a function similar to the denominator. This hints at using substitution for partial fractions.

Perform substitution

Let \( u = x^2 \). Then, \( du = 2x \, dx \) or \( x \, dx = \frac{1}{2} \, du \). Substitute these into the integral:\[\int \frac{x}{x^4 + 4} \, dx = \frac{1}{2} \int \frac{1}{u^2 + 4} \, du\]

Use trigonometric substitution

Recognize \( u^2 + 4 \) as resembling the identity for trigonometric substitution. Let \( u = 2 \tan(\theta) \), then \( du = 2 \sec^2(\theta) \, d\theta \). The integral becomes:\[\frac{1}{2} \int \frac{2 \sec^2(\theta)}{4 \tan^2(\theta) + 4} \, d\theta = \int \frac{\sec^2(\theta)}{\sec^2(\theta)} \, d\theta = \int d\theta = \theta\]

Reverse substitution

Since \( \theta = \arctan\left(\frac{u}{2}\right) \) and \( u = x^2 \), we have \( \theta = \arctan\left(\frac{x^2}{2}\right) \). The integral solves to:\[\theta + C = \arctan\left(\frac{x^2}{2}\right) + C\]

Final result

The integrated result of \( \int \frac{x}{x^4 + 4} \, dx \) is:\[\arctan\left(\frac{x^2}{2}\right) + C\]

Key concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify integration problems. It involves changing the variable to make the function easier to integrate. This technique is especially helpful when you spot a derivative of a function's part in the integral.
In the exercise provided, the expression \( \int \frac{x}{x^{4}+4} \, dx \) can be confusing at first glance. However, noticing that the numerator \(x\) resembles the derivative of \( x^2 \), which relates to the denominator, hints at using substitution. This method aims to simplify the integration process by transforming the variable.
Here’s how you typically apply the substitution method:

• Identify a substitution: Look for a part of the integral whose derivative is present elsewhere in the integrand. In this case, letting \( u = x^2 \) makes sense because the derivative \( du = 2x \, dx \) closely relates to the numerator \( x \, dx \).
• Replace and integrate: Substitute \( u \) into the integral and simplify it, making it easier to manage.
• Don't forget to revert substitutions after integrating to match the original variable.

This method can often make integration straightforward by reducing complex expressions into simpler ones.

Partial Fraction Decomposition

Partial fraction decomposition is another handy technique mainly used for integrating rational functions, where a polynomial is divided by another. The key idea is to break down complicated fractions into a sum of simpler fractions that are easier to integrate.
However, in the given exercise, the integral \( \int \frac{x}{x^{4}+4} \, dx \) benefits more directly from substitution rather than partial fraction decomposition. But recognizing the possibility of partial fractions is crucial. Note that sometimes, if the denominator can be factored, partial fraction decomposition is applicable.
This method involves:

• Decomposing the fraction: Express the fraction as a sum of simpler fractions, often using algebraic manipulation.
• Solving for constants: Determine the constants in the decomposed expression by equating coefficients or using values that simplify the expressions.
• Integrating each part separately: Once decomposed, integrate each simpler fraction individually, which is much easier than tackling the original complex fraction.

This technique is foundational in calculus for handling integrals involving rational functions.

Trigonometric Substitution

Trigonometric substitution is a technique applied when dealing with integrals involving square roots of quadratic expressions. By substituting trigonometric identities, it simplifies the integral based on trigonometric properties. This is especially useful for expressions with forms such as \( a^2 + x^2 \), \( a^2 - x^2 \), or \( x^2 - a^2 \).
In the exercise \( \int \frac{x}{x^4 + 4} \, dx \), trigonometric substitution comes into play after an initial variable substitution. Here, noticing that \( u^2 + 4 \) resembles a trigonometric identity, the substitution \( u = 2 \tan(\theta) \) is applied.
The core aspects of using trigonometric substitution include:

• Identifying the form: Recognize which trigonometric identity is akin to the expression in the integral.
• Performing substitution: Use an appropriate trigonometric function to replace \( u \). For example, if \( u^2 + 4 \), using \( u = 2 \tan(\theta) \) is efficient.
• Integrating with trigonometric identities: The integral often simplifies with trigonometric functions, allowing easier computation.
• Reverting the substitution: Always translate back to the original variable to complete the integration.

Trigonometric substitution is a powerful tool to tackle integrals with complex algebraic expressions by cleverly utilizing trigonometric identities.