integral of x(8x+5)^8

Lösung

\int x (8 x + 5)^{8} d x

\frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9}) + C

Schritte zur Lösung

\int x (8 x + 5)^{8} d x

Wende U-Substitution an

= \int \frac{u ^{8} ( u - 5 )}{64} d u

Entferne die Konstante:

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

= \frac{1}{64} \cdot \int u^{8} (u - 5) d u

Multipliziere aus

u^{8} (u - 5) : u^{9} - 5 u^{8}

= \frac{1}{64} \cdot \int u^{9} - 5 u^{8} 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}{64} (\int u^{9} d u - \int 5 u^{8} d u)

\int u^{9} d u = \frac{u ^{10}}{10}

\int 5 u^{8} d u = \frac{5 u ^{9}}{9}

= \frac{1}{64} (\frac{u ^{10}}{10} - \frac{5 u ^{9}}{9})

Setze in

u = 8 x + 5

ein

= \frac{1}{64} (\frac{( 8 x + 5 ) ^{10}}{10} - \frac{5 ( 8 x + 5 ) ^{9}}{9})

Vereinfache

\frac{1}{64} (\frac{( 8 x + 5 ) ^{10}}{10} - \frac{5 ( 8 x + 5 ) ^{9}}{9}) : \frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9})

= \frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9}) + C

\int x - \frac{1}{2} d x

y^{^{''}} + 3 y^{^{'}} + 2 y = \frac{1}{2 + e ^{x}}

t an g e n t f (x) = (1 + 4 x)^{\frac{3}{2}}, at x = 2

d er i v a t i v e y = x^{2} - 5 x - 3

\frac{d}{d x} (\frac{tan ^{2} ( x )}{2})