Problem 4 - University Calculus: Early Transcendentals

Question

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{4 x^{3}}{\left(x^{4}+1\right)^{2}} d x, u=x^{4}+1$$

Step-by-Step Solution

Answer

The solution is \(-\frac{1}{x^4 + 1} + C\).

Step 1: Substitute and Simplify

We are given the substitution \( u = x^4 + 1 \). First, find the derivative \( du \). Calculate \( du/dx \) which is \( 4x^3 \), and hence \( du = 4x^3 \, dx \). Substitute \( 4x^3 dx \) with \( du \). The integral becomes \( \int \frac{du}{u^2} \).

Step 2: Recognize the Integral Form

The integral \( \int \frac{du}{u^2} \) is now in the form of a standard integral \( \int u^{-2} \, du \).

Step 3: Integrate Using the Power Rule

Apply the power rule for integration to evaluate \( \int u^{-2} \, du \). According to the power rule, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = -2 \), so \( \int u^{-2} \, du = \frac{u^{-1}}{-1} + C = -\frac{1}{u} + C \).

Step 4: Back-Substitute for Original Variable

Substitute back \( u = x^4 + 1 \) into the expression \(-\frac{1}{u} + C \) to convert it back in terms of \( x \). Thus, the solution is \(-\frac{1}{x^4 + 1} + C \).

Key Concepts

Substitution Method

The substitution method is a technique used to simplify integrals, especially when a direct integration is challenging. It's like changing the setting of a problem to make it easier. In this method, you choose a substitution that simplifies the integral into a basic form to solve easily. For example, in the given problem, we use the substitution \( u = x^4 + 1 \). This choice is strategic because the derivative of \( x^4 + 1 \), which is \( 4x^3 \), appears in the numerator of the integral.

• First, identify a part of the integral that can be replaced by a simpler expression.
• Find the derivative of the substitution term \( u \) concerning \( x \), ensuring it matches the differential component of the integral.
• Replace all occurrences accordingly to express the integral solely in terms of \( u \).

After substitution, the integral becomes much more straightforward: \( \int \frac{du}{u^2} \). This transformed integral can now be approached using standard integration techniques.

Power Rule for Integration

The power rule for integration is one of the most basic yet powerful tools in calculus. It allows us to find the integral of power functions efficiently. According to the power rule, if you have an integral in the form \( \int u^n \, du \), the solution is \( \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \).

• This rule derives directly from reversing the process of differentiation for power functions.
• However, it doesn't apply when \( n = -1 \), as it leads to the natural logarithmic function instead.

In the exercise, once you have substituted correctly, realizing the integral form \( \int u^{-2} \, du \), you identify \( n = -2 \). Using the power rule, the integration becomes \( -\frac{1}{u} + C \), where \( C \) is the constant of integration.

Standard Integral Forms

Standard integral forms are pre-established integral solutions for common types of integrands. They form the backbone of solving many integral problems efficiently. When integrals are expressed in these forms, solving becomes a matter of direct application.

• They provide a shortcut by avoiding complex algebraic manipulations or substitutions that might otherwise be needed.
• Recognizing standard forms is essential, as it speeds up solving integrals, especially in exams or tests.

In our example, the integral \( \int \frac{1}{u^2} \, du \) fits the standard form \( \int u^{-2} \, du \). Recognizing this form allows us to directly apply the power rule without further alterations, leading to the solution \( -\frac{1}{u} + C \). Always ensure to substitute back the original variable at the end if a substitution was used, to express the solution in the initial context.