Question

For the following exercises, integrate using whatever method you choose.\(\int \frac{\sqrt{3+16 x^{4}}}{x^{4}} d x\)

Short answer

Use substitution with \( u = 3 + 16x^4 \) to simplify and integrate.

Step-by-step solution

Simplify the integrand

The integrand is given by \( \frac{\sqrt{3+16 x^{4}}}{x^{4}} \). Let's rewrite it as \( \sqrt{3+16x^4} \cdot x^{-4} \). This form will make it easier to handle in subsequent steps.

Simplify by substitution

To simplify the integration, use substitution. Let \( u = 3 + 16x^4 \), which means \( du = 64x^3 dx \). Solving for \( x^3 dx \) gives \( x^3 dx = \frac{1}{64} du \). Thus, \( dx = \frac{1}{64x^3} du \).

Change variables in the integral

By substitution, the integral becomes \( \int \frac{u^{1/2}}{x^4} \frac{1}{64x^3} \, du \). Rewrite \( x \) in terms of \( u \): since \( u = 3 + 16x^4 \), \( x^4 = \frac{u-3}{16} \). Hence, integrate \( \frac{u^{1/2}}{64 x^7} du \) by substituting \( x^4 \).

Solve the integral

Since \( x^4 = \frac{u-3}{16} \), it follows that \( x^7 = x^3 \times x^4 = \left( \frac{u - 3}{16} \right)^{7/4} \). Now, substitute into the integral: \[ \int \frac{u^{1/2}}{64 \left( \frac{u-3}{16} \right)^{7/4}} \, du = \frac{1}{64} \times 16^{7/4} \int u^{1/2} (u-3)^{-7/4} \, du \].

Evaluate the integral

The calculation simplifies as \( \frac{1}{64} \times 16^{7/4} \) is a constant which can be evaluated. After further simplification, the integral of \( u^{1/2} (u-3)^{-7/4} \) can be attempted using advanced integration techniques such as partial fraction decomposition or numerical methods for the remaining non-elementary integral.

Back substitution and finalize

After solving the integral, substitute back the original variable \( x \) using \( u = 3 + 16x^4 \). This gives the antiderivative in terms of \( x \).

Key concepts

Substitution Method

The substitution method, also known as the u-substitution in calculus, is a technique to simplify complex integrals by changing variables. It's especially useful when dealing with integrals where direct integration isn't straightforward. In our original exercise, we aim to make the integrand simpler to handle by choosing an appropriate substitution that transforms the integral into a more familiar form. To use the substitution method effectively:

• Identify a part of the integrand that, when replaced, simplifies the expression significantly. Often, this involves components of the integrand that when differentiated, resemble another part of the integrand.
• Define a new variable, say, \( u \), as this identified part. In the given problem, \( u = 3 + 16x^4 \) was chosen.
• Compute \( du \), the derivative of \( u \) with respect to \( x \), and solve it in terms of \( dx \). This step bridges the original variables and the new variable, facilitating a complete change of variables in the integral.
• Substitute \( u \) and \( du \) into the original integral, changing the limits if it's a definite integral, converting the original complex integral into a simpler one involving only \( u \).

Through substitution, we transformed the original integral into one involving \( u \, \), simplifying calculations and allowing us to focus on integrating a more manageable expression.

Integrand Simplification

Simplifying the integrand before integration can make the process significantly easier, particularly when dealing with complex expressions. In many calculus problems, including our example, initial expressions can look daunting. However, rewriting these equations can reveal simpler forms that are more straightforward to integrate. Steps to simplify the integrand can include:

• Breaking down components of the integrand such that complex expressions become manageable. As seen in the problem, \( \frac{\sqrt{3+16x^4}}{x^4} \) was rewritten as \( \sqrt{3+16x^4} \cdot x^{-4} \), separating the root and the fraction.
• Exploring identities or algebraic manipulations to reduce the integrand into recognizable patterns that are easier to handle. For example, rewriting exponentials can often open up opportunities for substitution or further simplification.
• Recognizing perfect squares or cube expressions which can be transformed using standard identities or rules.

By simplifying an integrand before solving, mathematical complexities are minimized, rendering subsequent steps like integration or substitution more efficient and less error-prone.

Complex Integrals

Complex integrals are integrals that require more advanced techniques beyond basic rules due to their complicated nature or the challenging form of the integrand. This problem exemplifies a complex integral where a direct antiderivative is not readily apparent, necessitating innovative approaches and possibly numerical methods. Key considerations when dealing with complex integrals include:

• Identifying appropriate methods or sequences of methods. Sometimes multiple techniques like substitution, integration by parts, or specialized numerical methods must be combined to handle particularly challenging integrals.
• Simplification and reduction strategies that transform the integral through algebraic manipulations and substitutions, as done with this problem.
• Utilizing or approximating non-elementary functions when standard elementary functions do not suffice. For example, after simplification, partial fraction decomposition or numerical approximations might become necessary.
• Understanding the underlying functions' behavior. This often involves examining asymptotic behavior or singularities which may affect the integration paths or solutions.

Handling complex integrals effectively often requires a mix of creativity and technical knowledge, but by following strategic simplification and methodical approaches, even the most difficult integrals become tractable.