https://file.aiquickdraw.com/mathbot/file/1728492912527f8upau5g.jpg

To solve the given integral using the indicated trigonometric substitution, we will follow these steps:

### Given Integral:
\[
\int \frac{dx}{x^2 \sqrt{4 - x^2}}
\]
with the substitution \( x = 2 \sin \theta \).

### Step 1: Trigonometric Substitution
The substitution \( x = 2 \sin \theta \) leads to the following:
- \( dx = 2 \cos \theta \, d\theta \)
- \( \sqrt{4 - x^2} = \sqrt{4 - 4\sin^2\theta} = \sqrt{4(1 - \sin^2\theta)} = 2\cos\theta \)

### Step 2: Substituting into the Integral
Substitute \( x = 2\sin\theta \), \( dx = 2\cos\theta\,d\theta \), and \( \sqrt{4 - x^2} = 2\cos\theta \) into the integral:

\[
\int \frac{dx}{x^2 \sqrt{4 - x^2}} = \int \frac{2 \cos \theta \, d\theta}{(2 \sin \theta)^2 \cdot 2 \cos \theta}
\]

This simplifies to:
\[
\int \frac{2 \cos \theta \, d\theta}{4 \sin^2 \theta \cdot 2 \cos \theta} = \int \frac{d\theta}{4 \sin^2 \theta}
\]
Simplifying further:
\[
\frac{1}{4} \int \csc^2 \theta \, d\theta
\]

### Step 3: Integrating
The integral of \( \csc^2 \theta \) is \( -\cot \theta \), so we have:
\[
\frac{1}{4} \left( -\cot \theta \right) + C = -\frac{1}{4} \cot \theta + C
\]

### Step 4: Back Substitution
We now need to express \( \cot \theta \) in terms of \( x \). From the substitution \( x = 2\sin\theta \), we know:
\[
\sin \theta = \frac{x}{2}
\]
Using the Pythagorean identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we get:
\[
\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \frac{x^2}{4}} = \frac{\sqrt{4 - x^2}}{2}
\]
So:
\[
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{4 - x^2}}{2}}{\frac{x}{2}} = \frac{\sqrt{4 - x^2}}{x}
\]

### Final Answer:
\[
-\frac{1}{4} \cot \theta + C = -\frac{1}{4} \cdot \frac{\sqrt{4 - x^2}}{x} + C = \frac{\sqrt{4 - x^2}}{4x} + C
\]

Thus, the evaluated integral is:
\[
\int \frac{dx}{x^2 \sqrt{4 - x^2}} = \frac{\sqrt{4 - x^2}}{4x} + C
\]

### Right Triangle Sketch:


Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. \( \int \frac{dx}{x^2 \sqrt{4 - x^2}} \) where \( x = 2 \sin \theta \)

Learn how to evaluate the integral \( \int \frac{dx}{x^2 \sqrt{4 - x^2}} \) using trigonometric substitution with the substitution \( x = 2 \sin \theta \). Step-by-step solution including a right triangle sketch and key formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation