Problem 58 - Calculus and its applications

Question

Evaluate. $$ \int_{-1}^{0} \frac{x^{3} d x}{\left(2-x^{4}\right)^{7}} $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{21}{512}\).

Step 1: Identify the integration method

The integral \( \int_{-1}^{0} \frac{x^{3} \, dx}{(2-x^{4})^{7}} \) involves a rational function that suggests a substitution might simplify the process. Here, substitution is likely the best approach.

Step 2: Consider Substitution

Let's use a substitution to simplify the integral. Define \( u = 2 - x^4 \). Then, the differential \( du = -4x^3 \, dx \). Rearranging gives \( x^3 \, dx = -\frac{1}{4} \, du \).

Step 3: Change Limits of Integration

With the substitution \( u = 2 - x^4 \), we need to change the integration limits. When \( x = -1 \), \( u = 2 - (-1)^4 = 1 \). When \( x = 0 \), \( u = 2 - 0^4 = 2 \). Thus, the new limits for \( u \) are from 1 to 2.

Step 4: Substitute and Integrate

Substituting into the integral gives \( \int_{u=1}^{u=2} -\frac{1}{4} \times \frac{1}{u^7} \, du \). This simplifies to \( -\frac{1}{4} \int_{1}^{2} u^{-7} \, du \).

Step 5: Solve the Integral

The integral \( \int u^{-7} \, du \) is evaluated as \( \frac{u^{-6}}{-6} + C = -\frac{1}{6u^6} + C \). So, \( -\frac{1}{4} \int_{1}^{2} u^{-7} \, du = \frac{1}{24} \left( \frac{1}{u^6} \right) \bigg|_1^2 \).

Step 6: Evaluate the Definite Integral

Substitute the limits into the evaluated integral: \( \frac{1}{24} \left( \frac{1}{2^6} - \frac{1}{1^6} \right) = \frac{1}{24} \left( \frac{1}{64} - 1 \right) = \frac{1}{24} \left( \frac{1}{64} - \frac{64}{64} \right) = \frac{1}{24} \times \frac{-63}{64} = -\frac{63}{1536} \).

Step 7: Simplify the Result

Simplify the fraction \( \frac{-63}{1536} \) by finding the greatest common divisor, which is 3. Thus, the simplified result is \( -\frac{21}{512} \).

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus, providing the total area under a curve within a specified range. Unlike indefinite integrals, which yield a family of functions, definite integrals give a specific numerical result, indicating accumulation of quantities like area or net change.
This is expressed using integral notation with upper and lower limits of integration, written as \[ \int_{a}^{b} f(x) \, dx \], where \( a \) and \( b \) are the bounds. A definite integral evaluates the function \( f(x) \) between these two points, \( a \) and \( b \).
To solve, you need to:

• Find an antiderivative or apply a substitution if necessary.
• Evaluate the antiderivative at both bounds.
• Subtract the result from the lower limit evaluation from the upper limit evaluation.

In practice, understanding the geometric interpretation aids in illuminating how integration conveys a summation of infinitely small parts across a continuous interval.

Rational Function

Rational functions are mathematical expressions involving ratios of polynomials. A general form is \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
Such functions get described by their numerators and denominators, dictating the shape and behavior of their graphs. Key features include:

• Vertical asymptotes, which occur where the denominator equals zero.
• Horizontal asymptotes or end behavior, based on the polynomials' degrees.
• Possible holes or discontinuities where factors in the numerator and denominator match.

In calculus, rational functions often require special techniques like substitution or partial fraction decomposition to integrate, particularly when the degrees vary. The integral in the exercise, \( \int \frac{x^3}{(2 - x^4)^7} \, dx \), presents a complex rational form, well-suited for integration by substitution given its composite structure.

Change of Variables

Change of variables, or substitution, is a strategic method in integration simplifying complex integrals by transforming them into a more manageable form. This involves substituting part of the integrand with a new variable, often denoted as \( u \), simplifying the integration process.
This method directly applies to the exercise by letting \( u = 2 - x^4 \), transforming the complex rational function into a simpler form. When performing a substitution, remember:

• Choose \( u \) such that its derivative DU appears in the integral.
• Adjust the differential, converting \( dx \) to \( du \), facilitating a direct integration.
• Change the integration bounds accordingly to \( u's \) new values, ensuring consistency.

This technique reduces the original integral into a form that's often easier to integrate, such as polynomial, exponential, or simple fraction. After integration, always revert back to the original variable to express the solution in its initial context.

Calculus Problem Solving

Solving calculus problems involves applying a structured approach, adapting methods to the problem's characteristics, and ensuring logical consistency throughout the process.
Key steps in problem solving include:

• Identifying the appropriate method of integration, like substitution, integration by parts, or using symmetry.
• Methodically performing algebraic manipulations, ensuring no steps are skipped or assumptions overlooked.
• Following through with accurate calculations, particularly in definite integrals, checking the limits and substitutions.
• Verifying the final result by simplifying or re-evaluating to confirm consistency with the problem's conditions.

Typically, calculus problems extend beyond computation. They require analyzing context, understanding underlying principles, and interpreting results meaningfully. Especially in technical fields, precise problem-solving in calculus forms the backbone of engineering, physics, and mathematics, fostering new solutions and innovations.