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

Lösung

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

\frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int (u - 1) u^{\frac{1}{2}} d u

Multipliziere aus

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

= \frac{1}{2} \cdot \int u^{\frac{3}{2}} - u^{\frac{1}{2}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int u^{\frac{1}{2}} d u = \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{2} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = x^{2} + 1

ein

= \frac{1}{2} (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}})

Vereinfache

\frac{1}{2} (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}) : \frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}}

= \frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}} + C

\int x^{2} sin (3 x^{3}) d x

\int e^{5 x - 8} d x

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

\int (tan (x))^{7} d x

\int sin^{7} (2 x) d x