integral of (x^3)/(x^2-36)

Lösung

\int \frac{x ^{3}}{x ^{2} - 36} d x

\frac{1}{2} (x^{2} - 36 + 36 ln x^{2} - 36) + C

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} - 36} d x

Wende U-Substitution an

= \int \frac{u + 36}{2 u} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u + 36}{u} d u

Multipliziere aus

\frac{u + 36}{u} : 1 + \frac{36}{u}

= \frac{1}{2} \cdot \int 1 + \frac{36}{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}{2} (\int 1 d u + \int \frac{36}{u} d u)

\int 1 d u = u

\int \frac{36}{u} d u = 36 ln ∣ u ∣

= \frac{1}{2} (u + 36 ln ∣ u ∣)

Setze in

u = x^{2} - 36

ein

= \frac{1}{2} (x^{2} - 36 + 36 ln x^{2} - 36)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x^{2} - 36 + 36 ln x^{2} - 36) + C

\int \frac{y}{3} d y

\int_{0}^{1} \frac{10 y}{e ^{2 y}} d y

\frac{\partial}{\partial y} (\frac{( a x + b y )}{c x + d y})

\int_{0}^{\infty} \frac{x}{x ^{3} + 1} d x

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