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Beliebt Rechnen >

integral of (x^3+x^2+5x+4)/((x^2+4)^2)

Lösung

\int \frac{x ^{3} + x ^{2} + 5 x + 4}{( x ^{2} + 4 ) ^{2}} d x

Lösung

- \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )} + C

Schritte zur Lösung

\int \frac{x ^{3} + x ^{2} + 5 x + 4}{( x ^{2} + 4 ) ^{2}} d x

Multipliziere aus

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

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 ^{3}}{( x ^{2} + 4 ) ^{2}} d x + \int \frac{x ^{2}}{( x ^{2} + 4 ) ^{2}} d x + \int \frac{5 x}{( x ^{2} + 4 ) ^{2}} d x + \int \frac{4}{( x ^{2} + 4 ) ^{2}} d x

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

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

\int \frac{5 x}{( x ^{2} + 4 ) ^{2}} d x = - \frac{5}{2 ( x ^{2} + 4 )}

\int \frac{4}{( x ^{2} + 4 ) ^{2}} d x = \frac{1}{4} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

= \frac{1}{2} (ln x^{2} + 4 + \frac{4}{x ^{2} + 4}) + \frac{1}{2} (\frac{1}{2} arctan (\frac{x}{2}) - \frac{x}{4 + x ^{2}}) - \frac{5}{2 ( x ^{2} + 4 )} + \frac{1}{4} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

Vereinfache

\frac{1}{2} (ln x^{2} + 4 + \frac{4}{x ^{2} + 4}) + \frac{1}{2} (\frac{1}{2} arctan (\frac{x}{2}) - \frac{x}{4 + x ^{2}}) - \frac{5}{2 ( x ^{2} + 4 )} + \frac{1}{4} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) : - \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )}

= - \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )} + C

Graph

Beliebte Beispiele

integral of (5x^2-x+6)/(x^3+2x-3)

\int \frac{5 x ^{2} - x + 6}{x ^{3} + 2 x - 3} d x

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\int x arcsin (\frac{x}{3}) d x

integral of 1/(x^2+8x+15)

\int \frac{1}{x ^{2} + 8 x + 15} d x

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\int \frac{7}{x ^{7}} d x

integral of (y+y^2)

\int (y + y^{2}) d y