Problem 30 - Applied Calculus

Question

Find each indefinite integral. \(\int(1-7 w) \sqrt[3]{w} d w\)

Step-by-Step Solution

Answer

\( \frac{3}{4}w^{4/3} - 3w^{7/3} + C \)

Step 1: Expand the Expression

Rewrite the integrand \( \int (1 - 7w) \sqrt[3]{w} \, dw \) as \( \int \left( \sqrt[3]{w} - 7w \cdot \sqrt[3]{w} \right) \, dw \). This is helpful for distributing the terms individually in the next steps.

Step 2: Express Roots as Powers

Express \( \sqrt[3]{w} \) as \( w^{1/3} \) and \( 7w \cdot \sqrt[3]{w} \) as \( 7w^{4/3} \). The integral becomes \[ \int \left( w^{1/3} - 7w^{4/3} \right) \, dw. \]

Step 3: Integrate Each Term Individually

Apply the power rule for integration, which states \( \int w^n \, dw = \frac{w^{n+1}}{n+1} + C \), to each term in the integrand.- For \( w^{1/3} \), integrate to get \( \frac{w^{4/3}}{4/3} = \frac{3}{4}w^{4/3} \).- For \( -7w^{4/3} \), integrate to get \( -7 \cdot \frac{w^{7/3}}{7/3} = -3w^{7/3} \).

Step 4: Combine Results and Simplify

Combine the results of the integrations: \( \frac{3}{4}w^{4/3} - 3w^{7/3} + C \),where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique used in calculus to find the indefinite integral of functions in the form of a power of a variable. The rule states that if you have an integral of the form \( \int x^n \, dx \), the result will be \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. This rule is applicable for any real number \( n \) except \( n = -1 \), as it would represent a logarithmic function.

When using the power rule, you increase the exponent \( n \) by one and divide by the new exponent. This simple rule makes it easier to handle polynomial and algebraic expressions. In practice, this means:\

• Identify each term in the integrand and express it as a power of the variable.
• Apply the power rule to each term individually.
• Don't forget to add the constant of integration \( C \) at the end, since the integration can have an infinite number of solutions.

Applying the power rule helps to efficiently find the integral of each term separately, simplifying complex expressions into manageable parts.

Integrating Algebraic Functions

Algebraic functions often consist of variables raised to a power, and integration of these functions involves applying rules like the power rule for integration. To integrate an algebraic function, follow some straightforward steps:

• Break down the function into simpler terms, if necessary. This often involves rewriting products or divisions to expose each term as a power.
• University expand and express roots or radicals as fractional exponents to better apply the power rule.
• Integrate each term individually using the power rule, which may require you to tailor each component of the expression to fit the \( x^n \) form.

Remember that in cases where you have coefficients, integrate as you would normally, keeping the coefficient outside the integral. Simplified expressions will let you handle these algebraic functions with sophistication, making the process an easily followed routine.

Cubic Roots in Integration

Cubic roots can appear in integrals as a challenge because they are not straightforward polynomial terms. To integrate expressions involving cubic roots, you need to represent the cube root in a form conducive to applying the power rule.

For example, the expression \( \sqrt[3]{w} \) can be rewritten as \( w^{1/3} \). This change is crucial for integration as it transforms the root into an exponent. Then, you can apply the power rule to find the integral:

• Rewrite the cubic root in the expression as a power \( x^{1/3} \).
• If the cubic root is part of a product, distribute it appropriately and express the entire term with exponents.
• Proceed with integration using the power rule, being mindful to adjust coefficients and term distribution as needed.

Understanding how to manipulate cubic roots and other roots into powers gives you the flexibility to tackle complex integrals with ease.