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Beliebt Rechnen >

integral from 1 to 1/x of cos^3(u)

Lösung

\int_{1}^{\frac{1}{x}} cos^{3} (u) d u

Lösung

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

Schritte zur Lösung

\int_{1}^{\frac{1}{x}} cos^{3} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{1}^{\frac{1}{x}} (1 - sin^{2} (u)) cos (u) d u

Wende U-Substitution an

= \int_{s i n (1)}^{s i n (\frac{1}{x})} 1 - v^{2} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{s i n (1)}^{s i n (\frac{1}{x})} 1 d v - \int_{s i n (1)}^{s i n (\frac{1}{x})} v^{2} d v

\int_{s i n (1)}^{s i n (\frac{1}{x})} 1 d v = sin (\frac{1}{x}) - sin (1)

\int_{s i n (1)}^{s i n (\frac{1}{x})} v^{2} d v = \frac{sin ^{3} ( \frac{1}{x} ) - sin ^{3} ( 1 )}{3}

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

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