Question

Use a table of integrals to evaluate the following integrals. $$ \int x^{3} \sqrt{1+2 x^{2}} d x $$

Short answer

Use substitution \(u = 1 + 2x^2\) to simplify and solve the integral step by step using the table's formulas.

Step-by-step solution

Identify the Integral Form

We need to evaluate the integral \( \int x^3 \sqrt{1+2x^2} \, dx \). First, compare this integral with known forms in a table of integrals; we're looking for one that matches or is similar so that substitutions can be made if necessary.

Select a Suitable Substitution

Given the integrand \(x^3 \sqrt{1+2x^2}\), notice the expression under the square root, \(1+2x^2\), suggests a substitution. Let \(u = 1+2x^2\). Then, \(du = 4x \, dx\), so \(x \, dx = \frac{1}{4} du\). Substitute to simplify the integral form.

Express \(x\) in Terms of \(u\)

From \(u = 1 + 2x^2\), solve for \(x^2 = \frac{u-1}{2}\) leading to \(x = \sqrt{\frac{u-1}{2}}\). Substituting \(x^2\) into the original wave, we have \((\sqrt{2(x^2)})^3 = \left(\frac{(u-1)}{2} \right)^{3/2}\).

Rewrite the Integral in Terms of \(u\)

Substitute \(x\), \(dx\), and the original term to express all parts in terms of \(u\): \(\int x^3 \sqrt{1+2x^2} \, dx = \int \left(\frac{u-1}{2}\right)^{3/2} \cdot \sqrt{u} \cdot \frac{1}{4} \frac{du}{2}\). This reduces to \(\frac{1}{8} \int (u-1)^{3/2} u^{1/2} \, du\).

Simplify the Integral Further

Combine and rearrange the powers: \(\frac{1}{8} \int (u-1)^{3/2} u^{1/2} \, du \). This becomes \(\frac{1}{8} \int u^2 (u-1)^{3/2} \, du \). Look up or derive this form from a standard integral table or use integration by parts.

Evaluate the Integral Using Known Results

Integrate \( \int u^2 (u-1)^{3/2} \ du \) using the integral substitution or parts, depending on your table. A typical method may involve expanding \((u-1)^{3/2}\) and integrating term by term.

Substitute Back to Original Variable

Once integration is complete, replace \(u\) with \(1+2x^2\) to revert the variable substitution. This provides the antiderivative in terms of \(x\).

Add the Constant of Integration

To finalize, express the integral result in terms of \(x\) and include the arbitrary constant \(C\) into your solution, as it's an indefinite integral.

Key concepts

Table of Integrals

When tackling complex integrals, like \( \int x^3 \sqrt{1+2x^2} \, dx \), a "Table of Integrals" can be incredibly handy. This table contains a variety of integral formulas and results derived from standard models. It serves as a reference tool for students and professionals alike, aiming to simplify the integration process.
By comparing the integrand you have with entries in the table, you might identify a similar form that helps solve the problem faster. This practice is particularly useful for integrals that don't initially appear straightforward or require substitution or transformation.
In our example, the complexity of \( x^3 \sqrt{1+2x^2} \) hints that a direct integration might be difficult. Instead, referring to a Table of Integrals can guide us to related standard forms that accommodate substitutions, providing a pathway to simplify and evaluate the integral effectively.

Substitution Method

The Substitution Method is essential in simplifying integrals involving complicated functions. It involves replacing a variable or expression in the integrand with another variable to simplify the integral evaluation.
In the integral \( \int x^3 \sqrt{1+2x^2} \, dx \), substituting \( u = 1+2x^2 \) considerably simplifies the process. Here’s how it works:

• Determine \( u \), leading to \( du = 4x \, dx \).
• Make the substitution \( x \, dx = \frac{1}{4} du \) and solve for \( x \).
• Rewrite the integral fully in terms of \( u \).

This approach transforms intricate integrals into more manageable forms, allowing for simpler computation. Substitution reduces the algebraic complexity and often leads to recognizable, solvable forms in the context of known integral tables.

Integration by Parts

Integration by Parts is another fundamental technique often employed when a substitution alone won't simplify the integrand to a standard form. It’s based on the product rule for differentiation and is particularly useful when dealing with products of functions.
The formula is stated as: \[ \int u \, dv = uv - \int v \, du \] This formula requires selecting parts of the integral to differentiate and integrate strategically. While our specific problem may not directly require Integration by Parts, complex forms derived from substitutions often see integrated simplification using this method.
For \( \int u^2 (u-1)^{3/2} \, du \), once substitution provides a suitable form, Integration by Parts or another round of substitution might be necessary. Employ intuition: choose \( u \) and \( dv \) such that \( du \) and \( v \) simplify the problem further towards eventual resolution.

Indefinite Integrals

Indefinite Integrals represent the process of finding a function whose derivative yields the given integrand. It is usually denoted without limits, focusing on identifying an antiderivative plus a constant \( C \).
For our integral \( \int x^3 \sqrt{1+2x^2} \, dx \), after simplifying through substitution and using techniques like integration by parts if needed, the result is expressed as an indefinite integral.
This approach forms the essential scaffolding of antiderivative calculation:

• Complete the integration process thoroughly.
• Substitute back any temporary variables used during substitution.
• Include the constant \( C \) to encompass the family of antiderivatives.

Indefinite integrals are vital in calculus as they form the groundwork for computing definite integrals and further applications involving rate of change and areas under curves. Always remember to add the constant of integration, ensuring the completeness of your solution.