integral of x(x^2+3)^{100}

Lösung

\int x (x^{2} + 3)^{100} d x

\frac{1}{202} (x^{2} + 3)^{101} + C

Schritte zur Lösung

\int x (x^{2} + 3)^{100} d x

Wende U-Substitution an

= \int \frac{u ^{100}}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int u^{100} d u

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{u ^{101}}{101}

Setze in

u = x^{2} + 3

ein

= \frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{101}}{101}

Vereinfache

\frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{101}}{101} : \frac{1}{202} (x^{2} + 3)^{101}

= \frac{1}{202} (x^{2} + 3)^{101}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{202} (x^{2} + 3)^{101} + C

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