https://file.aiquickdraw.com/mathbot/file/1730054505299trgzpjl1.jpg

Let's solve this step-by-step.

### Given:
We need to find the anti-derivative (indefinite integral):
\[
\int \frac{\sqrt{25 - x^2}}{x^2} \, dx
\]
with the hint: \(x = 5\sin\theta\).

---

### Step 1: Substitution
Given \(x = 5\sin\theta\), we can calculate the derivative:
\[
\frac{dx}{d\theta} = 5\cos\theta \quad \implies \quad dx = 5\cos\theta \, d\theta
\]
Also:
\[
\sqrt{25 - x^2} = \sqrt{25 - 25\sin^2\theta} = \sqrt{25(1 - \sin^2\theta)} = \sqrt{25\cos^2\theta} = 5\cos\theta
\]

---

### Step 2: Rewrite the integral using the substitution
Replacing \(x\) with \(5\sin\theta\) and \(dx\) with \(5\cos\theta \, d\theta\), the integral becomes:
\[
\int \frac{5\cos\theta}{(5\sin\theta)^2} \cdot 5\cos\theta \, d\theta
\]
Simplify:
\[
\int \frac{25\cos^2\theta}{25\sin^2\theta} \, d\theta = \int \frac{\cos^2\theta}{\sin^2\theta} \, d\theta = \int \cot^2\theta \, d\theta
\]

---

### Step 3: Integrate \( \cot^2\theta \)
We use the identity:
\[
\cot^2\theta = \csc^2\theta - 1
\]
Thus:
\[
\int \cot^2\theta \, d\theta = \int (\csc^2\theta - 1) \, d\theta = -\cot\theta - \theta + C
\]

---

### Step 4: Back-substitute \(\theta\)
Since \(x = 5\sin\theta\), we have:
\[
\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\sqrt{25 - x^2}}{x}
\]
Thus, the final answer becomes:
\[
-\cot\theta + \theta + C
\]

---

### Step 5: Verify the Correct Option
From the options given in the image, the correct answer is:
\[
\boxed{B: -\cot\theta + \theta + C}
\]

---

Would you like a deeper breakdown of any step?

---

**Related Questions:**
1. What are the key trigonometric identities useful in calculus?
2. How do you choose suitable substitutions for trigonometric integrals?
3. What is the integral of \(\csc^2\theta\) and where does it commonly appear?
4. How can one transform definite integrals when using trigonometric substitutions?
5. What are other common forms of integrals that use \(x = a\sin\theta\) substitutions?

**Tip:** Always remember to revert to the original variable when solving indefinite integrals with substitutions, unless you're asked to leave the answer in terms of \(\theta\).

The anti-derivative of the indefinite integral \( \int \frac{\sqrt{25 - x^2}}{x^2} \, dx \) with the hint: \( x = 5 \sin \theta \).

Learn how to find the anti-derivative of the indefinite integral \( \int \frac{\sqrt{25 - x^2}}{x^2} \, dx \) using trigonometric substitution. This solution involves applying the hint \( x = 5 \sin \theta \) and key trigonometric identities to solve for the correct answer. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation