integral of ((sin(x))/x)^2

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

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

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

\int \frac{cos ^{2} ( x )}{x ^{2}} d x = - \frac{cos ^{2} ( x )}{x} - Si (2 x)

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

y^{^{''}} - 12 y^{^{'}} + 37 y = 0, y (0) = 6, y^{^{'}} (0) = 8

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