Question

∫(2x/3).(√(4-x²) dx qual o resultado desta integral

Answer 1

Considerando que $4-x^2=u$, temos que:

$\frac{du}{dx}=-2x\therefore dx=-\frac{1}{2x}\;du$

Substituindo $dx$ na integral:

$\int \frac{2x}{3}\sqrt{4-x^2}\;dx=\int \frac{2x}{3}\sqrt{u}\cdot(-\frac{1}{2x})\;du$

$\int \frac{2x}{3}\sqrt{4-x^2}\;dx=\int-\frac{1}{3}\sqrt{u}\;du$

$\int \frac{2x}{3}\sqrt{4-x^2}\;dx=-\frac{1}{3}\int\sqrt{u}\;du$

$\int \frac{2x}{3}\sqrt{4-x^2}\;dx=-\frac{1}{3}\cdot \frac{2u^{3/2}}{3}+C$

$\int \frac{2x}{3}\sqrt{4-x^2}\;dx=-\frac{2u^{3/2}}{9}+C$

$\int \frac{2x}{3}\sqrt{4-x^2}\;dx=-\frac{2(4-x^2)^{3/2}}{9}+C$