Math Problem Statement
Solution
Let's analyze the problem and identify the correct basic integration formula and the value for .
The given integral:
We are asked to select a basic integration formula and identify .
Step 1: Simplify the expression
The term inside the square root is . Let's set up the substitution to reduce it to a standard form.
We notice the presence of a square root . Let's attempt a substitution , which means and .
Step 2: Apply the substitution
Substitute into the integral:
Now the expression becomes:
The integral becomes:
This simplifies to:
Step 3: Identify the correct formula
Now the integral is of the form:
where and corresponds to . This is a standard integral, and the basic integration formula we need is:
Thus, the correct formula from the options is:
Step 4: Identify
From the substitution earlier, we chose . Therefore, the correct identification is:
Would you like further details or clarification? Here are some related questions for further understanding:
- How do you apply trigonometric substitution in integrals?
- What are common substitution strategies for integrals involving square roots?
- How can the arcsine function arise from integrals of this form?
- What are some other cases where substitution simplifies integrals?
- How does one identify the right substitution technique for different integrals?
Tip: When simplifying integrals, always look for patterns involving basic identities or common substitution strategies to reduce complexity.
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Math Problem Analysis
Mathematical Concepts
Substitution in Integration
Definite and Indefinite Integrals
Square Root Function
Formulas
∫ du / √(a^2 - u^2)
u = √x
Theorems
Substitution Method
Standard Integral Forms
Suitable Grade Level
College level or Advanced Calculus
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