integral from 0 to 1 of x^3sqrt(1+3x^2)

Lösung

\int_{0}^{1} x^{3} 1 + 3 x^{2} d x

\frac{58}{135}

Dezimale

0.42962 \dots

Schritte zur Lösung

\int_{0}^{1} x^{3} 1 + 3 x^{2} d x

Wende U-Substitution an

= \int_{1}^{4} \frac{u ( u - 1 )}{18} d u

Entferne die Konstante:

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

= \frac{1}{18} \cdot \int_{1}^{4} u (u - 1) d u

Multipliziere aus

u (u - 1) : u^{\frac{3}{2}} - u

= \frac{1}{18} \cdot \int_{1}^{4} u^{\frac{3}{2}} - u 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}{18} (\int_{1}^{4} u^{\frac{3}{2}} d u - \int_{1}^{4} u d u)

\int_{1}^{4} u^{\frac{3}{2}} d u = \frac{62}{5}

\int_{1}^{4} u d u = \frac{14}{3}

= \frac{1}{18} (\frac{62}{5} - \frac{14}{3})

Vereinfache

\frac{1}{18} (\frac{62}{5} - \frac{14}{3}) : \frac{58}{135}

= \frac{58}{135}

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