integral of (4-x^2)^{3/2}

Lösung

\int (4 - x^{2})^{\frac{3}{2}} d x

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

Schritte zur Lösung

\int (4 - x^{2})^{\frac{3}{2}} d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d y}{d t} = 5 y

\int x + x d x

x \to \infty lim (\frac{e ^{2 x}}{e ^{2 x} + 1})

d er i v a t i v e f (x) = e^{8}

\frac{\partial}{\partial y} ((x e^{y})^{2})