Problem 1
Question
Find the integral. $$ \int \frac{5}{\sqrt{9-x^{2}}} d x $$
Step-by-Step Solution
Verified Answer
The integral of the given function is \(5 \frac{x}{3} + C\).
1Step 1: Trigonometric Substitution
Make the substitution \(x = 3 \sin{\theta}\) to simplify the expression under the square root. Calculate the differential \(dx = 3 \cos{\theta} d\theta\).
2Step 2: Expression Simplification
Substitute the value of \(x\) and \(dx\) into the integral, so it becomes \(\int \frac{5} {\sqrt{9-9\sin^{2}{\theta}}} * 3\cos{\theta} d\theta\). Simplify this expression to get \(\int 5\cos{\theta} d\theta\).
3Step 3: Integration
Integrate the expression \(\int5\cos{\theta} d\theta\) to get the integral in terms of \(\theta\). The integral of \(\cos{\theta}\) is \(\sin{\theta}\), so the result is \(5\sin{\theta} + C\), where \(C\) is the integration constant.
4Step 4: Back Substitution
Replace \(\theta\) with \(x\) using the relation \(x = 3 \sin{\theta}\), we get \(\theta = \arcsin(\frac{x}{3})\). Therefore, the final answer is \(5 \sin(\arcsin(\frac{x}{3})) + C = 5 \frac{x}{3} + C\).
Other exercises in this chapter
Problem 1
In Exercises \(1-6,\) evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\sinh 3\) (b) \(\tanh (-2)\
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In Exercises 1 and 2, use Example 1 as a model to evaluate the limit $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}$$ over the r
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Graphical Reasoning In Exercises \(1-4,\) use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, n
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Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}
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