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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

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

= \int \frac{x sin ( \frac{1}{x} ) - 1 + x}{x ^{3}} d x

Multipliziere aus

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

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

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

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

= cos (\frac{1}{x}) - (- \frac{1}{2 x ^{2}}) - \frac{1}{x}

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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