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Magazine Chapter 5: Problem 30
Find each indefinite integral. $$ \int(1-7 w) \sqrt[3]{w} d w $$
Short Answer
Expert verified
\( \frac{3}{4}w^{4/3} - 3w^{7/3} + C \).
Step by step solution
01
Distribute the Cubed Root
Distribute the cube root across the terms in the parenthesis, yielding two separate terms inside the integral: \[ \int ((1) \cdot w^{1/3} - (7w) \cdot w^{1/3}) \, dw = \int (w^{1/3} - 7w^{4/3}) \, dw \]
02
Integrate Each Term Separately
Now, integrate each term separately. Apply the power rule for integration: The power rule for integration states if \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).For \( w^{1/3} \), \( \int w^{1/3} \, dw = \frac{w^{1/3 + 1}}{1/3 + 1} = \frac{w^{4/3}}{4/3} \).For \( -7w^{4/3} \), \( \int -7w^{4/3} \, dw = -7 \cdot \frac{w^{4/3 + 1}}{4/3 + 1} = -7 \cdot \frac{w^{7/3}}{7/3} \).
03
Simplify Each Term
Simplify the results from each integration.For \( \frac{w^{4/3}}{4/3} \), it simplifies to \( \frac{3}{4}w^{4/3} \).For \( -7 \cdot \frac{w^{7/3}}{7/3} \), multiply by the reciprocal of \( 7/3 \) to obtain \( -3w^{7/3} \).
04
Compile the Final Antiderivative
Combine the simplified terms and remember to add the constant of integration \( C \):\[ \int (w^{1/3} - 7w^{4/3}) \, dw = \frac{3}{4}w^{4/3} - 3w^{7/3} + C \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule for Integration
The power rule for integration is a fundamental tool when finding the antiderivative of a function. This rule specifically applies to monomials, which are terms of the form \( x^n \). The beauty of the power rule is how it allows us to reverse the process of differentiation. When you see an expression like \( \int x^n \, dx \), you can directly apply the power rule to say:
- If \( n eq -1 \), the antiderivative is \( \frac{x^{n+1}}{n+1} \).
- The formula reflects the principle that the process of integration is the opposite operation of taking the derivative.
- Add 1 to the exponent \( n \).
- Divide by the new exponent \( n+1 \).
- This gives you the integral of \( x^n \) with respect to \( x \), adjusted by any coefficients in front of \( x^n \).
Antiderivative
The antiderivative is the reverse process of differentiation, meaning if you find the derivative of a function and then integrate it, you'll generally end up back at the original function. In mathematical terms:
- If \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
- It represents the family of all possible functions whose derivative is \( f(x) \).
- When you take \( \int w^{1/3} \, dw \), its antiderivative is given by \( \frac{3}{4}w^{4/3} \).
- The same goes for other terms, where each derivative is undone by integration.
Constant of Integration
Adding a constant of integration \( C \) is crucial when performing indefinite integrations. Whenever you integrate, this constant accounts for any constant term that could have been lost during differentiation.Here's how it plays a role:
- An indefinite integral represents a family of functions.
- Each member of this family differs by a constant value, which is where \( C \) comes in.
- This is because the derivative of a constant is zero, thus without \( C \), you're missing information from the original function.
- After performing indefinite integration, always remember: the general solution will include \(+ C\).
- For example, if your result of integration is \( \frac{3}{4}w^{4/3} - 3w^{7/3} \), align it with the constant by expressing it as \( \frac{3}{4}w^{4/3} - 3w^{7/3} + C \).
