To solve the integral 

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

we can proceed by using trigonometric substitution.

### Step 1: Use the substitution
Let \( x = \sin \theta \). Then, \( dx = \cos \theta \, d\theta \), and the integral becomes:

\[
\int \frac{1}{(1+\sin^2 \theta)\sqrt{1-\sin^2 \theta}} \cos \theta \, d\theta.
\]

### Step 2: Simplify the expression
Since \( \sqrt{1 - \sin^2 \theta} = \cos \theta \), the integral simplifies to:

\[
\int \frac{\cos \theta}{(1+\sin^2 \theta)\cos \theta} \, d\theta.
\]

This reduces further to:

\[
\int \frac{1}{1 + \sin^2 \theta} \, d\theta.
\]

### Step 3: Use the Weierstrass substitution or identity
We can use the identity \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \) to rewrite the integrand:

\[
\int \frac{1}{1 + \frac{1 - \cos(2\theta)}{2}} \, d\theta = \int \frac{1}{\frac{3 + \cos(2\theta)}{2}} \, d\theta.
\]

This simplifies to:

\[
\int \frac{2}{3 + \cos(2\theta)} \, d\theta.
\]

### Step 4: Solve the integral
The integral \( \int \frac{1}{a + b\cos(2\theta)} \, d\theta \) is a standard form. Using the formula:

\[
\int \frac{d

Learn how to solve the integral of 1 / ((1 + x^2) sqrt(1 - x^2)) dx with a detailed step-by-step solution using trigonometric substitution. This problem covers integral calculus and trigonometric identities, suitable for undergraduate students.

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