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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

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

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

\int 2 d x = 2 x

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

= \frac{1}{2} (x - \frac{1}{2} sin (2 x)) + 2 x - cot (x)

Vereinfache

\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + 2 x - cot (x) : \frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x)

= \frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x)

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x) + C

Graph

Beliebte Beispiele

integral of x/3 e^{-x/3}

\int \frac{x}{3} e^{- \frac{x}{3}} d x

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\int \frac{3 x ^{2} - 2 x + 19}{( x - 1 ) ( x ^{2} + 9 )} d x

integral of sin^2(8n-3)

\int sin^{2} (8 n - 3) d n

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\int (5 e^{x} + \frac{10}{x} - 4 x^{3} + 2^{x}) d x

integral of sin^2(x)*cos^7(x)

\int sin^{2} (x) \cdot cos^{7} (x) d x