integral of 2x\sqrt[4]{36+x^2}

Lösung

\int 2 x 4 36 + x^{2} d x

x^{2} 4 36 + x^{2} - \frac{1}{5} x^{2} 4 x^{2} + 36 + \frac{144}{5} 4 x^{2} + 36 + C

Schritte zur Lösung

\int 2 x 4 36 + x^{2} d x

Entferne die Konstante:

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

= 2 \cdot \int x 4 36 + x^{2} d x

Wende die partielle Integration an

= 2 (\frac{1}{2} x^{2} 4 36 + x^{2} - \int \frac{x ^{3}}{4 ( x ^{2} + 36 ) ^{\frac{3}{4}}} d x)

\int \frac{x ^{3}}{4 ( x ^{2} + 36 ) ^{\frac{3}{4}}} d x = \frac{1}{10} (x^{2} - 144) (x^{2} + 36)^{\frac{1}{4}}

= 2 (\frac{1}{2} x^{2} 4 36 + x^{2} - \frac{1}{10} (x^{2} - 144) (x^{2} + 36)^{\frac{1}{4}})

Vereinfache

2 (\frac{1}{2} x^{2} 4 36 + x^{2} - \frac{1}{10} (x^{2} - 144) (x^{2} + 36)^{\frac{1}{4}}) : x^{2} 4 36 + x^{2} - \frac{1}{5} x^{2} 4 x^{2} + 36 + \frac{144}{5} 4 x^{2} + 36

= x^{2} 4 36 + x^{2} - \frac{1}{5} x^{2} 4 x^{2} + 36 + \frac{144}{5} 4 x^{2} + 36

Füge eine Konstante zur Lösung hinzu

= x^{2} 4 36 + x^{2} - \frac{1}{5} x^{2} 4 x^{2} + 36 + \frac{144}{5} 4 x^{2} + 36 + C

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