find the indefinite integral. ∫x√4 - x² dx ∫x√4 - x² dx = □

Explanation:

Step1: Use substitution

Let ( u = 4 - x^{2} ), then ( du=-2x dx), and (x dx=-\frac{1}{2}du).

Step2: Substitute into the integral

The integral (\int x\sqrt{4 - x^{2}}dx) becomes (\int\sqrt{u}\left(-\frac{1}{2}\right)du=-\frac{1}{2}\int u^{\frac{1}{2}}du).

Step3: Integrate using power rule

Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (-\frac{1}{2}\times\frac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C=-\frac{1}{2}\times\frac{u^{\frac{3}{2}}}{\frac{3}{2}}+C).

Step4: Simplify and back - substitute

Simplify (-\frac{1}{2}\times\frac{2}{3}u^{\frac{3}{2}}+C=-\frac{1}{3}u^{\frac{3}{2}}+C). Substitute back (u = 4 - x^{2}), we get (-\frac{1}{3}(4 - x^{2})^{\frac{3}{2}}+C).

Answer:

(-\frac{1}{3}(4 - x^{2})^{\frac{3}{2}}+C)