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Beliebt Rechnen >

integral of (6+x^5)^25x^4

Lösung

\int (6 + x^{5})^{2} 5 x^{4} d x

Lösung

36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3} + C

Schritte zur Lösung

\int (6 + x^{5})^{2} \cdot 5 x^{4} d x

Entferne die Konstante:

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

= 5 \cdot \int (6 + x^{5})^{2} x^{4} d x

Wende U-Substitution an

= 5 \cdot \int \frac{( 6 + u ) ^{2}}{5} d u

Entferne die Konstante:

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

= 5 \cdot \frac{1}{5} \cdot \int (6 + u)^{2} d u

Multipliziere aus

(6 + u)^{2} : 36 + 12 u + u^{2}

= 5 \cdot \frac{1}{5} \cdot \int 36 + 12 u + u^{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

= 5 \cdot \frac{1}{5} (\int 36 d u + \int 12 u d u + \int u^{2} d u)

\int 36 d u = 36 u

\int 12 u d u = 6 u^{2}

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

= 5 \cdot \frac{1}{5} (36 u + 6 u^{2} + \frac{u ^{3}}{3})

Setze in

u = x^{5}

ein

= 5 \cdot \frac{1}{5} (36 x^{5} + 6 (x^{5})^{2} + \frac{( x ^{5} ) ^{3}}{3})

Vereinfache

5 \cdot \frac{1}{5} (36 x^{5} + 6 (x^{5})^{2} + \frac{( x ^{5} ) ^{3}}{3}) : 36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3}

= 36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3}

Füge eine Konstante zur Lösung hinzu

= 36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3} + C

Graph

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