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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

Beliebte Beispiele

integral from-1 to 4 of (x^2+2x-1)

\int_{- 1}^{4} (x^{2} + 2 x - 1) d x

derivative of-yx

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

49y^{''}+140y^'+100y=0

49 y^{^{''}} + 140 y^{^{'}} + 100 y = 0

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

\int (3 - x^{2}) 2^{x} d x

tangent ln(xy)+y^2=2y,(e,1)

t an g e n t ln (x y) + y^{2} = 2 y, (e, 1)