integral of x^3(1-x^2)^{-1/2}

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

x^{3} (1 - x^{2})^{- \frac{1}{2}} : \frac{x ^{3}}{( 1 - x ^{2} ) ^{\frac{1}{2}}}

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

Wende die partielle Integration an

= x^{3} arcsin (x) - \int 3 x^{2} arcsin (x) d x

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\int e^{- 2 \cdot x} d x

\int \frac{x}{x ^{2} + 2500} d x

\int \frac{5}{3 - 2 x} d x

\int a e^{- x} d x

\int x^{2} sin (x) cos (d) x d x