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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x ^{2} + 1} d x = arctan (x)

\int \frac{1}{( x ^{2} + 1 ) ^{2}} d x = \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative of-(sin(ln(x))/x)

\frac{d}{d x} (- \frac{sin ( ln ( x ) )}{x})

derivative of-x^2y

\frac{d}{d x} (- x^{2} y)

derivative sqrt(3-6x)

d er i v a t i v e 3 - 6 x

y^{''}-2y^'+y=6e^x

y^{^{''}} - 2 y^{^{'}} + y = 6 e^{x}

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h \to 0 lim (\frac{3 ^{h} - 1}{h})