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Beliebt Rechnen >

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

Lösung

\int \frac{8 - x}{( x ^{2} + 1 ) ^{2}} d x

Lösung

4 arctan (x) + \frac{4 x}{x ^{2} + 1} + \frac{1}{2 ( x ^{2} + 1 )} + C

Schritte zur Lösung

\int \frac{8 - x}{( x ^{2} + 1 ) ^{2}} d x

Wende die partielle Integration an

= \frac{1}{2} (8 - x) (arctan (x) + \frac{x}{1 + x ^{2}}) - \int - \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) d x

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

= \frac{1}{2} (8 - x) (arctan (x) + \frac{x}{1 + x ^{2}}) - (- \frac{1}{2} (x (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{1}{x ^{2} + 1}))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= 4 arctan (x) + \frac{4 x}{x ^{2} + 1} + \frac{1}{2 ( x ^{2} + 1 )} + C

Graph

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