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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{5}{2} arctan (x) + \frac{x}{2 ( 1 + x ^{2} )} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Vereinfache

2 arctan (x) + \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) : \frac{5}{2} arctan (x) + \frac{x}{2 ( 1 + x ^{2} )}

= \frac{5}{2} arctan (x) + \frac{x}{2 ( 1 + x ^{2} )}

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} arctan (x) + \frac{x}{2 ( 1 + x ^{2} )} + C

Graph

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