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Beliebt Rechnen >

integral of x(2x+3)^{99}

Lösung

\int x (2 x + 3)^{99} d x

Lösung

\frac{1}{4} (\frac{( 2 x + 3 ) ^{101}}{101} - \frac{3 ( 2 x + 3 ) ^{100}}{100}) + C

Schritte zur Lösung

\int x (2 x + 3)^{99} d x

Wende U-Substitution an

= \int \frac{u ^{99} ( u - 3 )}{4} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int u^{99} (u - 3) d u

Multipliziere aus

u^{99} (u - 3) : u^{100} - 3 u^{99}

= \frac{1}{4} \cdot \int u^{100} - 3 u^{99} 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}{4} (\int u^{100} d u - \int 3 u^{99} d u)

\int u^{100} d u = \frac{u ^{101}}{101}

\int 3 u^{99} d u = \frac{3 u ^{100}}{100}

= \frac{1}{4} (\frac{u ^{101}}{101} - \frac{3 u ^{100}}{100})

Setze in

u = 2 x + 3

ein

= \frac{1}{4} (\frac{( 2 x + 3 ) ^{101}}{101} - \frac{3 ( 2 x + 3 ) ^{100}}{100})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (\frac{( 2 x + 3 ) ^{101}}{101} - \frac{3 ( 2 x + 3 ) ^{100}}{100}) + C

Graph

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