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Beliebt Rechnen >

integral of (x^5+2x^3-3x)/((x^2+1)^3)

Lösung

\int \frac{x ^{5} + 2 x ^{3} - 3 x}{( x ^{2} + 1 ) ^{3}} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{5} + 2 x ^{3} - 3 x}{( x ^{2} + 1 ) ^{3}} d x

Ermittle den Partialbruch von

\frac{x ^{5} + 2 x ^{3} - 3 x}{( x ^{2} + 1 ) ^{3}} : \frac{x}{x ^{2} + 1} - \frac{4 x}{( x ^{2} + 1 ) ^{3}}

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 2e^{y^2}

\int 2 e^{y^{2}} d y

derivative e^{2x^x}

d er i v a t i v e e^{2 x^{x}}

integral from 1 to 4 of (x+sqrt(x))

\int_{1}^{4} (x + x) d x

integral of ((x^4+81))/(x(x^2+9)^2)

\int \frac{( x ^{4} + 81 )}{x ( x ^{2} + 9 ) ^{2}} d x

inverselaplace 1/(s^2-16)

in v erse l a pl a ce \frac{1}{s ^{2} - 16}