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Magazine Chapter 6: Problem 15
Apply Trigonometric Substitution to evaluate the indefinite integrals. \(\int \frac{3}{\sqrt{7-x^{2}}} d x\)
Short Answer
Expert verified
\( 3 \arcsin \left( \frac{x}{\sqrt{7}} \right) + C \).
Step by step solution
01
Identify the Trigonometric Substitution
The integral involves a square root of the form \( \sqrt{7-x^2} \), which is similar to \( \sqrt{a^2-x^2} \). For this, we use the trigonometric substitution \( x = a \sin \theta \). Here, \( a = \sqrt{7} \), so we substitute \( x = \sqrt{7} \sin \theta \).
02
Substitute and Simplify the Integral
We substitute \( x = \sqrt{7} \sin \theta \). The differential \( dx = \sqrt{7} \cos \theta \, d\theta \). Under this substitution, \( \sqrt{7-x^2} = \sqrt{7\cos^2 \theta} = \sqrt{7} \cos \theta \). Substitute these into the integral:\[\int \frac{3}{\sqrt{7-x^2}} \, dx = \int \frac{3}{\sqrt{7} \cos \theta} \cdot \sqrt{7} \cos \theta \, d\theta = \int 3 \, d\theta.\]
03
Integrate with Respect to \( \theta \)
Now the integral simplifies to \( \int 3 \, d\theta \). Integrating, we get:\[3\theta + C,\]where \( C \) is the constant of integration.
04
Back-Substitute for \( \theta \)
Back-substitute \( \theta \) using the relation \( x = \sqrt{7} \sin \theta \). So, \( \sin \theta = \frac{x}{\sqrt{7}} \). Thus, \( \theta = \arcsin \left( \frac{x}{\sqrt{7}} \right) \). Substitute this back to get:\[3 \arcsin \left( \frac{x}{\sqrt{7}} \right) + C.\]
05
Final expression for the indefinite integral
The expression \( 3 \arcsin \left( \frac{x}{\sqrt{7}} \right) + C \) represents the indefinite integral. Therefore, the solution is complete.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Indefinite Integrals
An indefinite integral represents a family of functions that describe the antiderivative of a given function. When you see the integral symbol \( \int \), it is asking you to find the original function before differentiation was applied.**Main Characteristics of Indefinite Integrals**:
- They include a constant of integration, \( C \), since differentiation of a constant results in zero.
- The process is synonymous with reversing differentiation. For example, the differentiation of \( x^2 \) gives you \( 2x \). Thus, the indefinite integral of \( 2x \) is \( x^2 + C \).
- They do not have limits of integration, distinguishing them from definite integrals which calculate area under a curve.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all angles. These identities are essential when working with integrals, especially when substitutions involve trigonometric expressions.**Commonly Used Identities**:
- \( \sin^2 \theta + \cos^2 \theta = 1 \) - Basic Pythagorean identity.
- \( 1 + \tan^2 \theta = \sec^2 \theta \) - Useful when dealing with tangent and secant functions.
- Trigonometric functions like sine, cosine, and tangent have reciprocal identities: \( \csc \theta = 1/\sin \theta \), \( \sec \theta = 1/\cos \theta \), \( \cot \theta = 1/\tan \theta \).
Integration Techniques
Integration techniques are methods used to simplify and solve integrals effectively. One of these techniques is trigonometric substitution, a powerful method when working with integrals involving quadratic expressions under a square root.**Why Trigonometric Substitution**:
- It simplifies the integral by converting it into a form that involves basic trigonometric integrals.
- Particularly useful for expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), and \( \sqrt{x^2 - a^2} \).
- Links the integrals to familiar trigonometric identities, enabling straightforward integration.
