integral of (11x^3)/(sqrt(2+x^2))

Lösung

\int \frac{11 x ^{3}}{2 + x ^{2}} d x

11 (\frac{1}{3} (2 + x^{2})^{\frac{3}{2}} - 2 2 + x^{2}) + C

Schritte zur Lösung

\int \frac{11 x ^{3}}{2 + x ^{2}} d x

Entferne die Konstante:

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

= 11 \cdot \int \frac{x ^{3}}{2 + x ^{2}} d x

Wende U-Substitution an

= 11 \cdot \int u^{2} - 2 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

= 11 (\int u^{2} d u - \int 2 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 2 d u = 2 u

= 11 (\frac{u ^{3}}{3} - 2 u)

Setze in

u = 2 + x^{2}

ein

= 11 (\frac{( 2 + x ^{2} ) ^{3}}{3} - 2 2 + x^{2})

Vereinfache

11 (\frac{( 2 + x ^{2} ) ^{3}}{3} - 2 2 + x^{2}) : 11 (\frac{1}{3} (2 + x^{2})^{\frac{3}{2}} - 2 2 + x^{2})

= 11 (\frac{1}{3} (2 + x^{2})^{\frac{3}{2}} - 2 2 + x^{2})

Füge eine Konstante zur Lösung hinzu

= 11 (\frac{1}{3} (2 + x^{2})^{\frac{3}{2}} - 2 2 + x^{2}) + C

\frac{d}{d x} (ln (x^{- 1}))

\frac{d}{d u} (tan (u))

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