https://r2.erweima.ai/mathbot/image/1725831977213tvh2vlrw.jpg

The given integral is:

\[
\int \frac{2x}{\sqrt{1 - x^2}} \, dx
\]

To solve this, let's follow the steps:

### Step 1: Identify a suitable substitution
Notice that the denominator involves \(\sqrt{1 - x^2}\), which suggests a trigonometric substitution. We can use:

\[
x = \sin(\theta)
\]

This leads to:
\[
dx = \cos(\theta) \, d\theta
\]
\[
\sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \cos(\theta)
\]

### Step 2: Substitute in terms of \(\theta\)
Substitute \(x = \sin(\theta)\) and \(dx = \cos(\theta) \, d\theta\) into the integral:

\[
\int \frac{2\sin(\theta)}{\cos(\theta)} \cdot \cos(\theta) \, d\theta
\]

Simplify the expression:
\[
\int 2\sin(\theta) \, d\theta
\]

### Step 3: Integrate
The integral of \(2\sin(\theta)\) is straightforward:

\[
2\int \sin(\theta) \, d\theta = -2\cos(\theta) + C
\]

### Step 4: Back-substitute \(\theta\)
Recall that \(x = \sin(\theta)\), which implies \(\cos(\theta) = \sqrt{1 - x^2}\). Therefore, the solution is:

\[
-2\sqrt{1 - x^2} + C
\]

Thus, the solution to the integral is:

\[
\boxed{-2\sqrt{1 - x^2} + C}
\]

---

Would you like a more detailed breakdown of any of the steps? 

Here are some additional related questions:
1. How do you choose which substitution to use in integrals like this one?
2. What is the general method for solving integrals involving square roots and squares?
3. How can trigonometric identities simplify more complex integrals?
4. What are other common trigonometric substitutions?
5. How does this solution change if a constant multiplier is added to the integrand?

**Tip:** Always look for patterns in integrals, such as the appearance of \(\sqrt{1 - x^2}\), which often signals a trigonometric substitution (like \(x = \sin(\theta)\)).

This page provides a step-by-step solution for evaluating the integral ∫(2x / √(1 - x²)) dx using trigonometric substitution. It covers important calculus concepts such as trigonometric identities and substitution methods. Suitable for undergraduate 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