Math Problem Statement
Solution
The given problem is to evaluate the definite integral:
Step-by-step Solution:
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Substitute: Let , so that .
- When , .
- When , .
Rewrite the integral in terms of :
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Simplify: The integral simplifies to:
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Integrate: Using the power rule , we have:
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Evaluate: Substitute the limits:
[ 4 \left[ u^{\frac{1}{2}} \right]_{1}^{4} = 4 \left[ \sqrt{4} - \sqrt{1} \right] = 4 \left[ 2 - 1 \right] = 4 \cdot 1 = 4 ]
Final Answer:
Do you want a detailed explanation of any step?
Related Questions:
- What is substitution in integration, and why is it useful?
- How does the power rule work for definite integrals?
- What is the geometric interpretation of this integral?
- How would the limits change if had a different range?
- What happens if the integral bounds are reversed (e.g., from to )?
Tip:
When integrating rational functions with square roots, substitution often simplifies the integral significantly!
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Math Problem Analysis
Mathematical Concepts
Definite Integrals
Substitution Method in Integration
Simplification of Rational Functions with Square Roots
Formulas
Substitution: Let \( u = g(x) \), \( du = g'(x) dx \).
Definite Integral with substitution: \( \int_a^b f(x) dx = \int_{g(a)}^{g(b)} f(u) \cdot \frac{du}{g'(x)} \)
Power Rule for Integration: \( \int u^n du = \frac{u^{n+1}}{n+1} + C \) for \( n \neq -1 \)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12 or early college-level calculus
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