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Magazine Chapter 11: Problem 3
Evaluate the indefinite integral. $$ \int \frac{\sqrt{4-x^{2}}}{x^{2}} d x $$
Short Answer
Expert verified
-\frac{1}{2} \frac{\sqrt{4-x^{2}}}{x} - \frac{1}{2} \arcsin{\frac{x}{2}} + C
Step by step solution
01
- Simplify the Integral
Rewrite the integrand to simplify later integration. Note that the integrand \( \frac{\sqrt{4-x^{2}}}{x^{2}} \) requires both algebraic manipulation and substitution.
02
- Use Trigonometric Substitution
Since the integral involves a square root of the form \( \sqrt{a^2 - x^2} \), use the substitution \( x = 2\sin{\theta} \). Then, compute \( dx \) and change the limits and the integrand accordingly. We have \( dx = 2\cos{\theta} d\theta \).
03
- Substitute and Simplify
Substitute \( x = 2\sin{\theta} \) into the integral. Hence the integrand becomes: \[ \int \frac{\sqrt{4 - (2\sin{\theta})^2}}{(2\sin{\theta})^2} \( 2\cos{\theta} d\theta \) \] Simplify: \[ \int \frac{\sqrt{4 - 4\sin^2{\theta}}}{4\sin^2{\theta}} \cdot 2\cos{\theta} d\theta = \int \frac{2\cos{\theta}}{4\sin^2{\theta}} \cdot 2\cos{\theta} d\theta = \int \frac{2\cos^2{\theta}}{2\sin^2{\theta}} d\theta \] Simplifying further: \[ \int \frac{\cos^2{\theta}}{2\sin^2{\theta}} d\theta = \frac{1}{2} \int \frac{\cos^2{\theta}}{\sin^2{\theta}} d\theta \]
04
- Use an Identity for Simplification
Recall the trigonometric identity: \( \cos^2{\theta} = 1 - \sin^2{\theta} \). Another helpful identity to transform the integrand is the Pythagorean identity: \( \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} \). So, reshape the integral: \[ \frac{1}{2} \int \frac{1-\sin^2{\theta}}{\sin^2{\theta}} d\theta = \frac{1}{2} \int \left( \frac{1}{\sin^2{\theta}} - 1 \right) d\theta = \frac{1}{2} \int \csc^2{\theta} d\theta - \frac{1}{2} \int d\theta \]
05
- Integrate
Now integrate each part separately. The integral of \( \csc^2{\theta} d\theta \) is \( -\cot{\theta} \) and the integral of \( d\theta \) is simply \( \theta \). Thus, we have: \[- \frac{1}{2} \cot{\theta} - \frac{1}{2} \theta + C \]
06
- Substitute Back
Revert the original substitution. Given that \( x = 2\sin{\theta} \), we can find \( \sin{\theta} = \frac{x}{2} \) and \( \theta = \arcsin{ \frac{x}{2} } \). Also, \( \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} \), so: \[ \cos{\theta} = \sqrt{1 - \left( \frac{x}{2} \right)^2} = \sqrt{ \frac{4 - x^2}{4} }. Thus, \cot{\theta} = \frac{ \sqrt{4 - x^2} }{x} \] Substituting these back, we get the final result: \[- \frac{1}{2} \cdot \frac{ \sqrt{4 - x^2} }{x} - \frac{1}{2} \arcsin{ \frac{x}{2} } + C \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Substitution
Trigonometric substitution is a powerful technique used when evaluating integrals involving square roots, where an expression resembles a trigonometric identity. In this case, the integral involves \( \sqrt{4 - x^2} \). When you see a term like this, a useful substitution is \( x = a \sin{\theta} \) or \( x = a \cos{\theta}\). For \( \sqrt{a^2 - x^2}\), using \( x = 2 \sin{\theta}\) simplifies the expression because \( \sin^2{\theta} + \cos^2{\theta} = 1\). Substitute back into the integral to transform it into a trigonometric integral.
Trigonometric Identities
Trigonometric identities play a crucial role in simplifying expressions within integrals. Remember the key trigonometric identities:
- The Pythagorean identity: \( \sin^2{\theta} + \cos^2{\theta} = 1 \)
- The quotient identity: \( \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\) and \( \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} \)
- The double-angle identities: \( \sin{2\theta} = 2\sin{\theta}\cos{\theta} \) and \( \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta} \)
Integral Simplification
Integral simplification is the process of transforming a complex integral into a form that is easier to integrate. Start by rewriting the original integral into a simpler form using algebraic manipulation or substitutions. For example, we changed \( \int \frac{\sqrt{4-x^2}}{x^2} dx \) by substituting \( x = 2\sin{\theta} \) to achieve a more straightforward trigonometric integral. Further simplification came from using trigonometric identities to handle terms like \( \int \csc^2{\theta} d\theta \). Integrate part by part and always remember to revert to the original variable post integration. This step-by-step simplification ensures the problem is broken down into more digestible parts, making complex integrals more approachable.
