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Chapter 5: Problem 54

Exer. \(53-56:\) Evaluate the integral by (a) the method of substitution and (b) expanding the integrand. In what way do the constants of integration differ? $$ \int\left(x^{2}+4\right)^{2} x d x $$

Short Answer

Expert verified
The integral is \( \frac{(x^2 + 4)^3}{6} + C_1 \) or \( \frac{x^6}{6} + 2x^4 + 8x^2 + C_2 \). Constants differ by a constant.

Step by step solution

01

Simplify the Integral Using Substitution

To use substitution, identify the composite function and set it equal to a new variable. Let \( u = x^2 + 4 \). The derivative of \( u \) with respect to \( x \) is \( du = 2x \, dx \), hence \( x \, dx = \frac{1}{2} du \). Substitute these into the integral:\[ \int (x^2 + 4)^2 x \, dx = \int u^2 \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^2 \, du \]
02

Integrate in Terms of Substitution Variable

Now, integrate with respect to \( u \):\[ \frac{1}{2} \int u^2 \, du = \frac{1}{2} \cdot \frac{u^3}{3} + C_1 = \frac{u^3}{6} + C_1 \]Substitute back \( u = x^2 + 4 \) to get the integral in terms of \( x \):\[ \frac{(x^2 + 4)^3}{6} + C_1 \]
03

Simplify the Integral by Expanding the Integrand

Rewrite the integrand by expanding \( (x^2 + 4)^2 \):\[ (x^2 + 4)^2 = (x^2)^2 + 2 \cdot x^2 \cdot 4 + 4^2 = x^4 + 8x^2 + 16 \]This changes the integral to:\[ \int (x^4 + 8x^2 + 16) x \, dx \]
04

Integrate the Expanded Function

Distribute \( x \) into the expanded polynomial:\[ \int (x^5 + 8x^3 + 16x) \, dx \]Integrate term by term:\[ \int x^5 \, dx = \frac{x^6}{6} \]\[ \int 8x^3 \, dx = 2x^4 \]\[ \int 16x \, dx = 8x^2 \]Combine the integrated terms:\[ \frac{x^6}{6} + 2x^4 + 8x^2 + C_2 \]
05

Compare Constants of Integration

In both parts, determine the constants of integration. The constants \( C_1 \) and \( C_2 \) may differ, as indefinite integrals always include an arbitrary constant. These constants account for any function vertically shifted by a constant amount.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Substitution Method
The substitution method is a helpful technique to simplify integral calculations. It can make complex expressions manageable by using a new variable to replace part of the integrand. In the problem provided, we let the expression \( x^2 + 4 \) equal \( u \). This simplifies the integrand considerably. Differentiating \( u \) with respect to \( x \) gives us \( du = 2x \, dx \). This allows the integral to be rewritten in terms of \( u \), making the computation simpler.
  • Finding the substitution: Identify a part of the integrand that, when replaced with a single variable, simplifies the calculation process.
  • Adjusting the differential: Substitute both the selected expression and its corresponding differential. For example, from \( x \, dx = \frac{1}{2} du \), we unify the expression in terms of \( du \).
  • Re-substitution: After integrating, substitute back the original expression to express the solution in terms of the original variable.
This method is powerful and reduces computational complexity, especially for problems with compositions like powers or roots.
What Is Integrand Expansion?
Integrand expansion involves rewriting a part of the integrand as a polynomial to simplify integration. In the problem example, we express \( (x^2 + 4)^2 \) as a polynomial. By expanding it into \( x^4 + 8x^2 + 16 \), each term becomes easier to integrate individually. This method is often used when integrals involve expressions raised to powers.
  • Expanding polynomials: Apply algebraic identities or direct multiplication to rewrite the expression.
  • Simplifying the integrand: Break down composite expressions into sums of monomials, making term-by-term integration easier.
  • Applying the integral: Integrate each term separately, which often allows using basic integration rules without additional substitution.
Integrand expansion simplifies specific forms and is especially useful when the expression aligns well with known algebraic identities or can be simplified directly.
The Importance of Constants of Integration
In indefinite integrals, a constant of integration is always present. This constant signifies the spectrum of possible vertical shifts of the anti-derivative function, meaning there are infinitely many solutions that differ by a constant.
  • Arbitrary constants: Denoted as \( C_1 \) or \( C_2 \), these constants accommodate the indefinite nature of indefinite integrals.
  • Components of solutions: Each method for solving an integral (e.g., substitution, expansion) might lead to different constants, but the solution's form matters more than the constant鈥檚 specific value.
  • Graphical interpretation: The constant affects the vertical position of the curve but not its shape.
Understanding the role of these constants is crucial, especially in physics or engineering, where the boundary or initial conditions determine specific values of these constants.

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