Problem 15 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{x^{3} d x}{\sqrt{x^{2}+4}} $$

Step-by-Step Solution

Answer

Use substitution \( u = x^2 + 4 \) and simplify; then integrate and back-substitute.

Step 1: Identify the Type of Integration

The integral \( \int \frac{x^{3} dx}{\sqrt{x^{2}+4}} \) involves a rational function where the denominator involves a square root. This suggests that substitution might simplify the integral.

Step 2: Select a Suitable Substitution

Let's perform the substitution \( u = x^2 + 4 \). Then the differential is \( du = 2x \, dx \), or equivalently, \( x \, dx = \frac{1}{2} \, du \). This substitution simplifies the denominator.

Step 3: Express x in Terms of u

Using the substitution \( u = x^2 + 4 \), it follows that \( x^2 = u - 4 \). Therefore, \( x^3 = x \cdot x^2 = x(u-4) \).

Step 4: Rewrite the Integral in Terms of u

Substitute all expressions in terms of \( u \):\[ \int \frac{x^{3} dx}{\sqrt{x^{2}+4}} = \int \frac{x(u-4) \frac{1}{2} du}{\sqrt{u}} \]Simplifying, we substitute \( x = \sqrt{u-4} \), giving:\[ \frac{1}{2} \int \frac{(u-4)^{\frac{1}{2}} (u-4) du}{u^{\frac{1}{2}}} \].

Step 5: Simplify the New Integral

Simplify the expression:\[ \frac{1}{2} \int \frac{(u-4) u - 4(u-4)}{u^{1/2}} du \]which becomes:\[ \frac{1}{2} \int (u^{1/2} (u-4)) - \frac{4(u-4)}{u^{1/2}} du \].

Step 6: Separate and Integrate

Break it into two separate integrals:\[ = \frac{1}{2} \left( \int u^{1/2}(u-4) \, du - 4\int \frac{(u-4)}{u^{1/2}} du \right) \]. Simplify each term and solve using standard integration techniques for each resulting polynomial.

Step 7: Back-Substitute to Original Variable

After integrating with respect to \( u \), substitute back \( u = x^2 + 4 \) to express the result in terms of \( x \). Add the constant of integration at this step.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in integral calculus, particularly useful when dealing with complex integrals. It involves replacing a variable in the integral with another variable to simplify the integration process.
This method is especially handy when you have a composition of functions, like a function inside another function. In our example, we substituted \( u = x^2 + 4 \) as this adequately simplifies the expression under the square root.
Here's why this substitution works well:

• By substituting \( u = x^2 + 4 \), the expression \( u \) simplifies the square root part and transforms the problem into a form based on \( u \), making it easier to integrate.
• The differential part transforms as well: \( du = 2x \, dx \), so \( x \, dx = \frac{1}{2} \, du \). This substitution allows us to rewrite and simplify the integral in terms of \( u \), eliminating the square root.

Using substitution is like changing the frame of reference of a problem to see it more clearly. It is a key skill for efficiently solving challenging integrals.

Rational Functions

Rational functions are quotients of two polynomials. When paired with square roots, they present unique challenges in calculus. Understanding rational functions is crucial when dealing with integrals, as they often require specific techniques like substitution or partial fractions to simplify.
In our exercise, we focus on integrating a function of the form:

• The numerator is a polynomial \( x^3 \), which can be handled straightforwardly.
• The denominator, however, involves \( \sqrt{x^2+4} \), complicating the process.

The complexity arises due to the mix of polynomial terms and the square root. The crafted substitution \( x^2 + 4 \) aims to streamline this obstacle. The transformed integral then becomes more approachable using known techniques for integrals of rational functions, particularly since it boils down to simpler polynomial terms in \( u \).
Accurate manipulation of rational functions can vastly simplify integrals, often reducing a seemingly difficult problem into a series of simpler steps.

Integration Techniques

Applying a variety of integration techniques is essential in solving complex integrals, especially when direct computation is not feasible. In our scenario, the integration required involves breaking down the integral into manageable parts after substitution.
Here are the steps we followed for effective integration:

• Simplified the integral post-substitution to \( \frac{1}{2} \int \left[ u^{1/2}(u-4) - \frac{4(u-4)}{u^{1/2}} \right] \ du \).
• Separated the integrated terms properly: This often means breaking a complex expression down into simpler terms which can be integrated using elementary techniques.
• Utilized polynomial multiplication and power rules for integration. Each term was simplified, integrated, and then all results were combined back into the original variable \( x \).
• Last, but not least, always remember to substitute back to the original variable and add a constant of integration to signify any potential constant term that could have originally been part of the function.

Mastering these steps and acknowledging the role of integration techniques in calculus will significantly enhance problem-solving capabilities and deepen understanding of complex integrals.