integral of x(4-x^2)^{1/2}

Lösung

\int x (4 - x^{2})^{\frac{1}{2}} d x

x (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - 2 (- \frac{1}{12} (4 - x^{2})^{\frac{3}{2}} + x arcsin (\frac{x}{2}) + 4 - x^{2}) + C

Schritte zur Lösung

\int x (4 - x^{2})^{\frac{1}{2}} d x

Wende die partielle Integration an

= x (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - \int 2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x)) d x

\int 2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x)) d x = 2 (- \frac{1}{12} (4 - x^{2})^{\frac{3}{2}} + x arcsin (\frac{x}{2}) + 4 - x^{2})

= x (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - 2 (- \frac{1}{12} (4 - x^{2})^{\frac{3}{2}} + x arcsin (\frac{x}{2}) + 4 - x^{2})

Füge eine Konstante zur Lösung hinzu

= x (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - 2 (- \frac{1}{12} (4 - x^{2})^{\frac{3}{2}} + x arcsin (\frac{x}{2}) + 4 - x^{2}) + C

\frac{\partial}{\partial y} (x y e^{x y})

\int \frac{x - x}{3} d x

x \to 0 lim (\frac{( x )}{x})

\int \frac{3 x + 1}{3 x - 1} d x

x \to 1 lim (\frac{3 x + 2 - 1}{x - 1})