sympy import symbols, integrate, sqrt

Define the variables

x = symbols('x')

Define the integral

expression = x / sqrt(1 - x**2)

Solve the integral

result = integrate(expression, x) resultThe solution to the integral

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

is:

$-\sqrt{1 - x^2} + C$

where $C$ is the constant of integration.

Would you like further clarification on the solution?

Here are 5 related questions to consider:

1. How do you derive this solution step-by-step?
2. What are common applications of such integrals in physics or engineering?
3. How does this solution relate to trigonometric substitution methods?
4. What would change if the integrand were slightly altered, such as $\sqrt{1 - x^2}$ in the numerator?
5. Can we verify this solution by differentiating it?

Tip: Always check your integration by differentiating the result to confirm correctness.