Math Problem Statement
integral of square root 16-x^2
Solution
The integral you're looking to solve is:
This is a standard integral that can be solved using a trigonometric substitution. Here's the step-by-step approach:
Step 1: Use Trigonometric Substitution
We will use the substitution , since resembles the identity .
Thus, let: Then,
Step 2: Simplify the Integrand
Substitute into the integrand:
So the integral becomes:
Step 3: Use a Trigonometric Identity
We can simplify using the half-angle identity:
So, the integral becomes:
Step 4: Integrate
Now, integrate:
Step 5: Reverse the Substitution
Recall that , so . Now, solve for using .
Also, , and from the Pythagorean identity .
Thus, .
Now substitute these into the integral result:
[ 8 \left[ \theta + \frac{\sin(2\theta)}{2} \right] = 8 \left[ \arcsin\left( \frac{x}{4} \right) + \frac{x \sqrt{16 - x^2}}{16} \right] + C ]
Final Answer:
Would you like further details or have any questions about this solution?
Related Questions:
- How does the trigonometric substitution method help simplify integrals involving square roots of quadratic expressions?
- Can you derive the integral of using a similar substitution method?
- What is the significance of the half-angle identity in reducing trigonometric integrals?
- How do you handle definite integrals with square roots of quadratic functions using trigonometric substitution?
- Can the result be applied to other integrals, like ?
Tip: Always remember that recognizing a standard form like can greatly simplify the process of integration. Trigonometric substitution is a powerful tool for this!
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Math Problem Analysis
Mathematical Concepts
Integration
Trigonometric Substitution
Trigonometric Identities
Formulas
cos^2(θ) = (1 + cos(2θ))/2
sin^2(θ) + cos^2(θ) = 1
Integration formula for √(a^2 - x^2): ∫√(a^2 - x^2) dx = (a^2/2)arcsin(x/a) + (x/2)√(a^2 - x^2) + C
Theorems
Pythagorean Identity
Trigonometric Substitution Theorem
Suitable Grade Level
Undergraduate Calculus (or advanced high school calculus)
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