integral from 0 to 4 of x^2sqrt(2x+1)

Lösung

\int_{0}^{4} x^{2} 2 x + 1 d x

\frac{5884}{105}

Dezimale

56.03809 \dots

Schritte zur Lösung

\int_{0}^{4} x^{2} 2 x + 1 d x

Wende U-Substitution an

= \int_{1}^{9} \frac{u ( u - 1 ) ^{2}}{8} d u

Entferne die Konstante:

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

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

Multipliziere aus

u (u - 1)^{2} : u^{\frac{5}{2}} - 2 u^{\frac{3}{2}} + u

= \frac{1}{8} \cdot \int_{1}^{9} u^{\frac{5}{2}} - 2 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}{8} (\int_{1}^{9} u^{\frac{5}{2}} d u - \int_{1}^{9} 2 u^{\frac{3}{2}} d u + \int_{1}^{9} u d u)

\int_{1}^{9} u^{\frac{5}{2}} d u = \frac{4372}{7}

\int_{1}^{9} 2 u^{\frac{3}{2}} d u = \frac{968}{5}

\int_{1}^{9} u d u = \frac{52}{3}

= \frac{1}{8} (\frac{4372}{7} - \frac{968}{5} + \frac{52}{3})

Vereinfache

\frac{1}{8} (\frac{4372}{7} - \frac{968}{5} + \frac{52}{3}) : \frac{5884}{105}

= \frac{5884}{105}

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