Math Problem Statement
Evaluate the indefinite integral: ∫ x^4/√1-x^10 dx
Solution
To evaluate the indefinite integral
Step-by-Step Solution:
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Substitution: Let . Then: Also, notice that .
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Rewriting the integral: Substituting and , the integral becomes:
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Simplifying: Factor out the constant :
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Recognizing the standard integral: The integral is a standard form, which equals .
Thus:
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Back-substitution: Recall that , so:
Final Answer:
Would you like further clarification on any of the steps?
Related Questions:
- How is the substitution method applied in integral problems?
- What is the derivation of the formula ?
- How does substitution help simplify integrals with powers and roots?
- Can you provide a geometrical interpretation of ?
- What are other techniques for solving integrals involving square roots?
Tip:
When encountering integrals with , consider substitution to simplify powers or recognize standard trigonometric forms.
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Math Problem Analysis
Mathematical Concepts
Integration
Substitution method
Trigonometric integrals
Formulas
∫ 1/√(1 - u^2) du = arcsin(u) + C
Substitution: Let u = g(x), then du = g'(x) dx
Theorems
Fundamental theorem of calculus
Standard integrals involving square roots
Suitable Grade Level
Grades 11-12
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