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Beliebt Rechnen >

integral of (1-x^2)/(x+x^3)

Lösung

\int \frac{1 - x ^{2}}{x + x ^{3}} d x

Lösung

ln ∣ x ∣ - ln x^{2} + 1 + C

Schritte zur Lösung

\int \frac{1 - x ^{2}}{x + x ^{3}} d x

Ermittle den Partialbruch von

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

= \int \frac{1}{x} - \frac{2 x}{x ^{2} + 1} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x} d x - \int \frac{2 x}{x ^{2} + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2 x}{x ^{2} + 1} d x = ln x^{2} + 1

= ln ∣ x ∣ - ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - ln x^{2} + 1 + C

Graph

Beliebte Beispiele

derivative of 5x^3+7x^{2.5}-3x+1

\frac{d}{d x} (5 x^{3} + 7 x^{2.5} - 3 x + 1)

integral of (4/x+x/4)

\int (\frac{4}{x} + \frac{x}{4}) d x

(\partial)/(\partial y)(5xy)

\frac{\partial}{\partial y} (5 x y)

ln(x)(dy)/(dx)= y/x

ln (x) \frac{d y}{d x} = \frac{y}{x}

(\partial)/(\partial y)(e^x+xcos(y))

\frac{\partial}{\partial y} (e^{x} + x cos (y))