To solve the integral $\int \frac{1}{\sqrt{4x - x^2}} \, dx,$ we’ll go through it step by step.

Step 1: Complete the Square

Rewrite the expression under the square root to make it easier to work with: $4x - x^2 = -(x^2 - 4x).$ To complete the square on $x^2 - 4x$, rewrite it as: $x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4.$ Thus, we can rewrite the integrand as: $\int \frac{1}{\sqrt{4 - (x - 2)^2}} \, dx.$

Step 2: Trigonometric Substitution

Since we have an expression of the form $4 - (x - 2)^2$, let’s use the substitution: $x - 2 = 2 \sin \theta.$ Then, $dx = 2 \cos \theta \, d\theta.$ Substituting into the integrand, we get: $\int \frac{1}{\sqrt{4 - 4 \sin^2 \theta}} \cdot 2 \cos \theta \, d\theta.$

Step 3: Simplify the Integral

Simplify the square root: $\sqrt{4 - 4 \sin^2 \theta} = \sqrt{4 (1 - \sin^2 \theta)} = \sqrt{4 \cos^2 \theta} = 2 \cos \theta.$ Substitute this into the integrand: $\int \frac{1}{2 \cos \theta} \cdot 2 \cos \theta \, d\theta = \int 1 \, d\theta = \theta + C.$

Step 4: Substitute Back

Since $x - 2 = 2 \sin \theta$, we have $\sin \theta = \frac{x - 2}{2}.$ Therefore, $\theta = \arcsin \left( \frac{x - 2}{2} \right).$ Substituting back, we get: $\int \frac{1}{\sqrt{4x - x^2}} \, dx = \arcsin \left( \frac{x - 2}{2} \right) + C.$

Final Answer

$\int \frac{1}{\sqrt{4x - x^2}} \, dx = \arcsin \left( \frac{x - 2}{2} \right) + C.$

Would you like further clarification on any of these steps?

Related Questions

1. How do you handle integrals with forms similar to $\int \frac{1}{\sqrt{a^2 - x^2}} \, dx$?
2. What is trigonometric substitution, and when is it most useful in integrals?
3. How can we check the answer of an integral by differentiating the result?
4. What are other methods for solving integrals involving square roots?
5. How can completing the square help with integrals outside of trigonometric substitution?

Tip

For integrals involving square roots of quadratic expressions, completing the square can simplify the expression, often making it suitable for trigonometric substitution.