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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (sqrt(25-16x^2))/x

\int \frac{25 - 16 x ^{2}}{x} d x

integral of 1/(sqrt(1-v^2))

\int \frac{1}{1 - v ^{2}} d v

integral of 1/(x^3(sqrt((x^2-9))))

\int \frac{1}{x ^{3} ( ( x ^{2} - 9 ) )} d x

derivative 1/(u^{1/2)}

d er i v a t i v e \frac{1}{u ^{\frac{1}{2}}}

tangent f(x)=2x^2-3x,\at x=4

t an g e n t f (x) = 2 x^{2} - 3 x, at x = 4