integral of x^2sqrt(8x-7)

Lösung

\int x^{2} 8 x - 7 d x

\frac{1}{3360} (120 x^{2} + 84 x + 49) (8 x - 7)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{2} 8 x - 7 d x

Wende U-Substitution an

= \int \frac{u ( u + 7 ) ^{2}}{512} d u

Entferne die Konstante:

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

= \frac{1}{512} \cdot \int u (u + 7)^{2} d u

Multipliziere aus

u (u + 7)^{2} : u^{\frac{5}{2}} + 14 u^{\frac{3}{2}} + 49 u

= \frac{1}{512} \cdot \int u^{\frac{5}{2}} + 14 u^{\frac{3}{2}} + 49 u 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}{512} (\int u^{\frac{5}{2}} d u + \int 14 u^{\frac{3}{2}} d u + \int 49 u d u)

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 14 u^{\frac{3}{2}} d u = \frac{28}{5} u^{\frac{5}{2}}

\int 49 u d u = \frac{98}{3} u^{\frac{3}{2}}

= \frac{1}{512} (\frac{2}{7} u^{\frac{7}{2}} + \frac{28}{5} u^{\frac{5}{2}} + \frac{98}{3} u^{\frac{3}{2}})

Setze in

u = 8 x - 7

ein

= \frac{1}{512} (\frac{2}{7} (8 x - 7)^{\frac{7}{2}} + \frac{28}{5} (8 x - 7)^{\frac{5}{2}} + \frac{98}{3} (8 x - 7)^{\frac{3}{2}})

Vereinfache

\frac{1}{512} (\frac{2}{7} (8 x - 7)^{\frac{7}{2}} + \frac{28}{5} (8 x - 7)^{\frac{5}{2}} + \frac{98}{3} (8 x - 7)^{\frac{3}{2}}) : \frac{1}{3360} (120 x^{2} + 84 x + 49) (8 x - 7)^{\frac{3}{2}}

= \frac{1}{3360} (120 x^{2} + 84 x + 49) (8 x - 7)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3360} (120 x^{2} + 84 x + 49) (8 x - 7)^{\frac{3}{2}} + C