integral of ((sqrt(x)+5)^9)/(2sqrt(x))

Lösung

\int \frac{( x + 5 ) ^{9}}{2 x} d x

x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9} + C

Schritte zur Lösung

\int \frac{( x + 5 ) ^{9}}{2 x} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{( x + 5 ) ^{9}}{x} d x

Wende U-Substitution an

= \frac{1}{2} \cdot \int 2 (u + 5)^{9} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot 2 \cdot \int (u + 5)^{9} d u

Wende die partielle Integration an

= \frac{1}{2} \cdot 2 ((u + 5)^{9} u - \int 9 (u + 5)^{8} u d u)

\int 9 (u + 5)^{8} u d u = \frac{9}{10} (u + 5)^{10} - 5 (u + 5)^{9}

= \frac{1}{2} \cdot 2 ((u + 5)^{9} u - (\frac{9}{10} (u + 5)^{10} - 5 (u + 5)^{9}))

Setze in

u = x

ein

= \frac{1}{2} \cdot 2 ((x + 5)^{9} x - (\frac{9}{10} (x + 5)^{10} - 5 (x + 5)^{9}))

Vereinfache

\frac{1}{2} \cdot 2 ((x + 5)^{9} x - (\frac{9}{10} (x + 5)^{10} - 5 (x + 5)^{9})) : x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9}

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

Füge eine Konstante zur Lösung hinzu

= x (x + 5)^{9} - \frac{9}{10} (x + 5)^{10} + 5 (x + 5)^{9} + C

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