Use the substitution  x=3tant,  −π2≤t≤π2x=3tan⁡t,  −π2≤t≤π2  to simplify the following integral:

∫19+x2dx∫19+x2dx
 

We are given the integral:

\[
\int \frac{1}{9 + x^2} \, dx
\]

and the substitution \( x = 3\tan t \), with \( -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} \).

### Step-by-step solution:

1. **Differentiate** the substitution \( x = 3\tan t \):
   \[
   dx = 3\sec^2 t \, dt
   \]

2. **Substitute** \( x = 3\tan t \) and \( dx = 3\sec^2 t \, dt \) into the integral:
   \[
   \int \frac{1}{9 + x^2} \, dx = \int \frac{1}{9 + (3\tan t)^2} \cdot 3\sec^2 t \, dt
   \]

3. Simplify the expression \( 9 + (3\tan t)^2 \):
   \[
   9 + (3\tan t)^2 = 9 + 9\tan^2 t = 9(1 + \tan^2 t)
   \]
   Using the trigonometric identity \( 1 + \tan^2 t = \sec^2 t \), we get:
   \[
   9(1 + \tan^2 t) = 9\sec^2 t
   \]

4. Substitute this back into the integral:
   \[
   \int \frac{1}{9\sec^2 t} \cdot 3\sec^2 t \, dt = \int \frac{3\sec^2 t}{9\sec^2 t} \, dt = \int \frac{1}{3} \, dt
   \]

5. Integrate with respect to \( t \):
   \[
   \int \frac{1}{3} \, dt = \frac{t}{3} + C
   \]

6. **Substitute** back \( t \) in terms of \( x \) using the inverse of \( x = 3\tan t \), which gives \( t = \tan^{-1}\left(\frac{x}{3}\right) \):
   \[
   \frac{t}{3} = \frac{1}{3} \tan^{-1} \left( \frac{x}{3} \right)
   \]

Thus, the simplified integral is:

\[
\int \frac{1}{9 + x^2} \, dx = \frac{1}{3} \tan^{-1} \left( \frac{x}{3} \right) + C
\]

### Would you like further details or clarification?

Here are 5 questions related to this process:
1. How does substitution simplify integrals involving squares of variables?
2. What is the significance of using \( \tan^{-1} \) in integrals?
3. Why do we restrict the bounds for \( t \) to \( -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} \)?
4. Can we use other trigonometric substitutions for similar integrals?
5. How do we handle definite integrals when performing substitutions?

**Tip:** Always check for trigonometric identities to simplify expressions when using substitutions.

Use the substitution x=3tan⁡t, −π/2≤t≤π/2 to simplify the following integral: ∫ 1/(9+x^2) dx

Learn how to simplify the integral ∫ 1/(9 + x^2) dx using the substitution x = 3tan t, and applying trigonometric identities. This step-by-step guide includes key concepts in integral calculus, trigonometric substitution, and inverse trigonometric functions. Suitable for advanced high school and college-level students.

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