https://file.aiquickdraw.com/mathbot/file/17310576376557waztvyw.jpg

To solve the integral:

\[
\int \frac{x}{\sqrt{4 - x^2}} \, dx
\]

we can proceed by using a substitution method. Here’s a step-by-step solution:

### Step 1: Use Trigonometric Substitution
Since the expression under the square root is of the form \(4 - x^2\), we can use the substitution:
\[
x = 2\sin \theta
\]
which implies that
\[
dx = 2\cos \theta \, d\theta
\]

Then, the expression \(4 - x^2\) becomes:
\[
4 - x^2 = 4 - 4\sin^2 \theta = 4\cos^2 \theta
\]
so that
\[
\sqrt{4 - x^2} = 2\cos \theta
\]

### Step 2: Substitute into the Integral
Now we can rewrite the integral in terms of \(\theta\):
\[
\int \frac{x}{\sqrt{4 - x^2}} \, dx = \int \frac{2\sin \theta}{2\cos \theta} \cdot 2\cos \theta \, d\theta
\]

Simplifying this expression:
\[
= \int 2\sin \theta \, d\theta
\]

### Step 3: Integrate with Respect to \(\theta\)
The integral now becomes:
\[
\int 2\sin \theta \, d\theta = -2\cos \theta + C
\]

### Step 4: Substitute Back in Terms of \(x\)
Since we originally set \(x = 2\sin \theta\), we can solve for \(\cos \theta\):
\[
\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{x}{2}\right)^2} = \frac{\sqrt{4 - x^2}}{2}
\]

Thus,
\[
-2\cos \theta = -\sqrt{4 - x^2}
\]

### Final Answer
The integral simplifies to:
\[
\int \frac{x}{\sqrt{4 - x^2}} \, dx = -\sqrt{4 - x^2} + C
\]

Would you like a deeper explanation on any of these steps?

Here are some additional questions you might find useful:
1. How does trigonometric substitution help in evaluating integrals with roots?
2. What other types of integrals can benefit from trigonometric substitution?
3. Can we evaluate this integral using a different substitution method?
4. How can we apply this method to similar integrals involving different constants?
5. What are the practical applications of solving integrals of this form?

**Tip:** For integrals involving \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), or \(\sqrt{x^2 - a^2}\), trigonometric substitutions like \(x = a \sin \theta\), \(x = a \tan \theta\), and \(x = a \sec \theta\) are often useful.

Evaluate the integral ∫ (x / √(4 - x^2)) dx.

This solution demonstrates how to evaluate the integral ∫ (x / √(4 - x^2)) dx by applying trigonometric substitution. Follow each step as we transform the integral using x = 2sin(θ), simplifying to a solvable integral. Ideal 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