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Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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![### Calculus Problem: Definite Integral Evaluation
**Problem:**
Evaluate the definite integral from 0 to 1 of the function \( \frac{dx}{x^3 + 3x^2} \).
\[ \int_{0}^{1} \frac{dx}{x^3 + 3x^2} \]
**Explanation:**
To solve this integral, one possible approach is to simplify the integrand function first. This involves factorizing the denominator and possibly using methods such as partial fraction decomposition if applicable.
### Steps to Solve:
1. **Factor the Denominator:**
\[ x^3 + 3x^2 = x^2(x + 3) \]
Here, \( x^3 + 3x^2 \) can be factorized into \( x^2(x + 3) \).
2. **Rewrite the Integral:**
\[ \int_{0}^{1} \frac{dx}{x^2(x + 3)} \]
3. **Use Partial Fraction Decomposition:**
Decompose:
\[ \frac{1}{x^2(x + 3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 3} \]
Determine constants A, B, C by solving the equation:
\[ 1 = A x(x + 3) + B(x + 3) + Cx^2 \]
4. **Integrate Each Term:**
Solve the integrals formed after decomposing into simpler fractions.
5. **Apply Limits:**
Calculate the definite integral by applying the limits 0 to 1 on the integrated results.
This method requires a detailed step-by-step approach to fully decompose, integrate, and apply the limits. Each step involves algebraic manipulation followed by integration techniques, ensuring to correctly handle the bounds of the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faaa612e4-bb6a-4a7c-9bcb-a4980796bdd1%2Fcd3e0f42-4827-4732-87f3-1962744fb2ae%2F54uudzk_reoriented.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 2 images
