https://file.aiquickdraw.com/mathbot/file/1725433632439py90m5jc.jpg

The integral presented in the image is:

\[
\int \frac{dx}{x\sqrt{9 - x^2}}
\]

You are asked to evaluate this integral using trigonometric substitution.

### Solution:

For integrals involving expressions of the form \(\sqrt{a^2 - x^2}\), the appropriate trigonometric substitution is:

\[
x = a \sin\theta
\]

Given the integral:

\[
\int \frac{dx}{x\sqrt{9 - x^2}}
\]

We can identify \(a = 3\) (since \(9 = 3^2\)). Thus, the substitution is:

\[
x = 3\sin\theta
\]

Differentiating with respect to \(\theta\):

\[
dx = 3\cos\theta \, d\theta
\]

Substituting \(x = 3\sin\theta\) and \(dx = 3\cos\theta \, d\theta\) into the integral:

\[
\int \frac{3\cos\theta \, d\theta}{3\sin\theta \cdot \sqrt{9 - 9\sin^2\theta}}
\]

Simplify the square root:

\[
\sqrt{9(1 - \sin^2\theta)} = \sqrt{9\cos^2\theta} = 3\cos\theta
\]

Substituting this back:

\[
\int \frac{3\cos\theta \, d\theta}{3\sin\theta \cdot 3\cos\theta} = \int \frac{d\theta}{\sin\theta}
\]

Now, the integral simplifies to:

\[
\int \csc\theta \, d\theta
\]

The integral of \(\csc\theta\) is:

\[
\int \csc\theta \, d\theta = \ln\left|\csc\theta - \cot\theta\right| + C
\]

Now, substituting back the original variable \(x\) using \(x = 3\sin\theta\):

\[
\sin\theta = \frac{x}{3}, \quad \cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \left(\frac{x}{3}\right)^2} = \frac{\sqrt{9 - x^2}}{3}
\]

Also, we know:

\[
\csc\theta = \frac{1}{\sin\theta} = \frac{3}{x}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\sqrt{9 - x^2}}{x}
\]

Thus, the solution is:

\[
\ln\left|\frac{3}{x} - \frac{\sqrt{9 - x^2}}{x}\right| + C = \ln\left|\frac{3 - \sqrt{9 - x^2}}{x}\right| + C
\]

This is the final answer:

\[
\ln\left|\frac{3 - \sqrt{9 - x^2}}{x}\right| + C
\]

Would you like further details or explanations? Here are some follow-up questions to deepen your understanding:

1. What is the general approach to solving integrals involving \(\sqrt{a^2 - x^2}\)?
2. How would the result differ if the integral had \(\sqrt{x^2 - 9}\) instead?
3. What are other possible substitutions for similar integrals?
4. How does the integral change if the constant in the square root was different (e.g., \(4 - x^2\))?
5. Why is it important to substitute back to the original variable after integration?

**Tip:** Trigonometric substitution is particularly useful when dealing with integrals that involve square roots of quadratic expressions. Understanding which substitution to use can significantly simplify the problem.

Evaluate the integral using trigonometric substitution: ∫(dx / x√(9 - x²))

This page provides a step-by-step solution to evaluate the integral ∫(dx / x√(9 - x²)) using trigonometric substitution. The problem explores key concepts in calculus, including trigonometric substitution and the integral of csc(θ). Suitable for university-level students studying calculus.

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