https://file.aiquickdraw.com/mathbot/file/172955148657422zm0x6n.jpg

Let’s evaluate the integral:

\[
I = \int_0^1 \sin^{-1}\left(\frac{2x}{1 + x^2}\right) \, dx
\]

We’ll solve this using **substitution**.

---

### Step 1: Substitution
We notice that:

\[
u = \frac{2x}{1 + x^2}
\]

Now, let's differentiate \(u\) with respect to \(x\).

\[
\frac{du}{dx} = \frac{(1 + x^2)(2) - 2x(2x)}{(1 + x^2)^2} = \frac{2 - 2x^2}{(1 + x^2)^2} = \frac{2(1 - x^2)}{(1 + x^2)^2}
\]

Thus,

\[
du = \frac{2(1 - x^2)}{(1 + x^2)^2} \, dx
\]

---

### Step 2: Adjust the Substitution  
We rewrite the term \( \sin^{-1}(u) \). However, this direct substitution gets slightly complicated for integration since the derivative of \( u \) is complex.  

Instead, observe that:

\[
\frac{2x}{1 + x^2} = \sin(2\theta) \quad \text{if} \quad x = \tan(\theta).
\]

Thus, a **trigonometric substitution** could be helpful. Let’s try the trigonometric substitution \( x = \tan(\theta) \), which transforms the integrand.

---

### Step 3: Trigonometric Substitution
With \( x = \tan(\theta) \), we have:

\[
dx = \sec^2(\theta) \, d\theta, \quad 1 + x^2 = \sec^2(\theta).
\]

Thus,

\[
\frac{2x}{1 + x^2} = \sin(2\theta).
\]

The integral becomes:

\[
I = \int_0^{\pi/4} \sin^{-1}(\sin(2\theta)) \, \sec^2(\theta) \, d\theta.
\]

Since \( \sin^{-1}(\sin(2\theta)) = 2\theta \) for \( 0 \leq \theta \leq \pi/4 \), the integral simplifies to:

\[
I = \int_0^{\pi/4} 2\theta \, \sec^2(\theta) \, d\theta.
\]

---

### Step 4: Evaluate the Integral
Evaluating this integral is now straightforward.

Using any suitable integration technique, evaluate the integral from 0 to 1 of sin^(-1)(2x / (1 + x^2)) dx, giving your answer in exact form.

Learn how to evaluate the integral from 0 to 1 of sin^(-1)(2x / (1 + x^2)) using substitution and trigonometric techniques. This problem covers key concepts in inverse trigonometric functions and advanced integration methods, 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