Problem 40 - Calculus

Question

Show that \(\int 2 x^{3} d x /\left(1+x^{2}\right)^{3}=\int(u-1) d u / u^{3}=\)

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{1}{2u} + \frac{1}{4u^2} + C'\).

Step 1: Identify the Substitution

We are given the integral \( \int \frac{2x^3}{(1+x^2)^3} \, dx \). Notice that \( 1 + x^2 \) is inside a function raised to a power. Let's use the substitution method: let \( u = 1 + x^2 \). Then \( du = 2x \, dx \). This substitution helps simplify our integral.

Step 2: Change Variables

Now, using the substitution \( u = 1 + x^2 \), we find \( du = 2x \, dx \). Thus, \( dx = \frac{du}{2x} \). Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{2x^3}{(1+x^2)^3} \, dx = \int \frac{x^3}{u^3} \cdot \frac{du}{2x} \].

Step 3: Simplify the Integral

In the integral \( \int \frac{x^3}{u^3} \cdot \frac{du}{2x} \), simplify by canceling \( x \) in the numerator and denominator, obtaining: \( \int \frac{x^2}{2u^3} \, du \). However, we need an expression entirely in terms of \( u \). Substitute back \( x^2 = u - 1 \), which yields \( \int \frac{u-1}{2u^3} \, du \).

Step 4: Split the Integral

Split the integrand into two separate terms: \( \int \left( \frac{u}{2u^3} - \frac{1}{2u^3} \right) \, du \). Simplifying each term gives \( \int \left( \frac{1}{2u^2} - \frac{1}{2u^3} \right) \, du \).

Step 5: Perform Integration

Integrate each term separately: \( \int \frac{1}{2u^2} \, du - \int \frac{1}{2u^3} \, du \). The integral \( \int u^{-2} \, du = -\frac{1}{u} + C \) and \( \int u^{-3} \, du = -\frac{1}{2u^2} + C \). Thus \[ \int \frac{1}{2u^2} \, du = -\frac{1}{2u} \quad \text{and} \quad \int \frac{1}{2u^3} \, du = -\frac{1}{4u^2} \].

Step 6: Combine Results

Combine the results from the previous step: \[-\frac{1}{2u} + \frac{1}{4u^2} + C' \]. This represents the integrated result of the original expression in terms of \( u \).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in calculus used to simplify integrals by changing variables. The main idea is to transform a complex integral into a simpler one by substituting a part of the integral with a new variable. This process often turns a difficult problem into something more manageable.

For instance, consider the integral \( \int \frac{2x^3}{(1+x^2)^3} \, dx \). By identifying a part of the integrand, such as \( 1 + x^2 \), we can set it equal to a new variable, say \( u \). This gives us the relationship:

• Let \( u = 1 + x^2 \)
• Then \( du = 2x \, dx \)

This substitution changes the integral's variable and reduces complexity. It allows us to express \( x^2 \) as \( u - 1 \), fully transitioning the integral to the \( u \) terms, simplifying calculation.

Definite Integrals

Definite integrals are used in calculus to calculate the area under a curve between two boundaries. Unlike indefinite integrals, which provide a general form with an arbitrary constant \( C \), definite integrals produce a precise number.Let's imagine we needed to calculate the area from one point to another for \( \int \frac{2x^3}{(1+x^2)^3} \, dx \). By setting the limits of integration before substitution, we'd transform the limits as well, ensuring they fit our new variable.

• The original bounds in terms of \( x \) become updated to reflect the change to \( u \).
• Incorporate the function's new form to calculate the exact area.

Substituting intelligently helps us keep the integrity of the calculation while simplifying the expression.

U-Substitution

U-substitution is a specific type of substitution method where the new variable is labeled as \( u \). It is especially useful when dealing with integrals involving functions within functions, such as composite functions. It simplifies the derivative and, consequently, the integral.In our exercise, by setting \( u = 1 + x^2 \), we see:

• \( du = 2x \, dx \)

From our substitution, \( dx \) can be rewritten as \( \frac{du}{2x} \). This change is pivotal as it transforms the integrand from one involving \( x \) into an entirely new expression involving \( u \):

• \( \int \frac{x^3}{u^3} \cdot \frac{du}{2x} \)

Through simplification and further substitutions, \( x^2 = u - 1 \), it continues to adjust the integral into a fully functioning \( u \)-context, allowing for straightforward integration.

Integration Techniques

Integration techniques are methods used to find integrals of various forms. They include methods like substitution, integration by parts, and partial fractions. Each technique is tailored to handle specific forms of functions or polynomials that can arise within an integrand.In our case, the substitution forms an essential technique, particularly for integrals with expressions raised to a power.

• By substituting, the integral transforms, paving the way for potential fraction division.
• The splitted integral \( \int \left( \frac{u}{2u^3} - \frac{1}{2u^3} \right) \, du \) becomes two separate integrals.

These integrations, \( \int \frac{1}{2u^2} \, du \) and \( \int \frac{1}{2u^3} \, du \), can then be solved individually using laws of exponents. It is vital to choose the integration technique matching the structure of the expression, ensuring an efficient route to a solution.