integral of (cos^4(x))/(sin^7(x))

Lösung

\int \frac{cos ^{4} ( x )}{sin ^{7} ( x )} d x

\frac{1}{384} (tan^{6} (\frac{x}{2}) - 3 tan^{4} (\frac{x}{2}) - 3 tan^{2} (\frac{x}{2}) + 24 ln tan (\frac{x}{2}) + 3 cot^{2} (\frac{x}{2}) + 3 cot^{4} (\frac{x}{2}) - cot^{6} (\frac{x}{2})) + C

Schritte zur Lösung

\int \frac{cos ^{4} ( x )}{sin ^{7} ( x )} d x

Wende U-Substitution an

= \int \frac{( - u ^{2} + 1 ) ^{4} ( u ^{2} + 1 ) ^{2}}{64 u ^{7}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{64} \cdot \int \frac{( - u ^{2} + 1 ) ^{4} ( u ^{2} + 1 ) ^{2}}{u ^{7}} d u

Lange Division

\frac{( - u ^{2} + 1 ) ^{4} ( u ^{2} + 1 ) ^{2}}{u ^{7}} : u^{5} - 2 u^{3} - u + \frac{4 u ^{6} - u ^{4} - 2 u ^{2} + 1}{u ^{7}}

= \frac{1}{64} \cdot \int u^{5} - 2 u^{3} - u + \frac{4 u ^{6} - u ^{4} - 2 u ^{2} + 1}{u ^{7}} d u

Wende die Summenregel an:

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

= \frac{1}{64} (\int u^{5} d u - \int 2 u^{3} d u - \int u d u + \int \frac{4 u ^{6} - u ^{4} - 2 u ^{2} + 1}{u ^{7}} d u)

\int u^{5} d u = \frac{u ^{6}}{6}

\int 2 u^{3} d u = \frac{u ^{4}}{2}

\int u d u = \frac{u ^{2}}{2}

\int \frac{4 u ^{6} - u ^{4} - 2 u ^{2} + 1}{u ^{7}} d u = 4 ln ∣ u ∣ + \frac{1}{2 u ^{2}} + \frac{1}{2 u ^{4}} - \frac{1}{6 u ^{6}}

= \frac{1}{64} (\frac{u ^{6}}{6} - \frac{u ^{4}}{2} - \frac{u ^{2}}{2} + 4 ln ∣ u ∣ + \frac{1}{2 u ^{2}} + \frac{1}{2 u ^{4}} - \frac{1}{6 u ^{6}})

Setze in

u = tan (\frac{x}{2})

ein

= \frac{1}{64} (\frac{tan ^{6} ( \frac{x}{2} )}{6} - \frac{tan ^{4} ( \frac{x}{2} )}{2} - \frac{tan ^{2} ( \frac{x}{2} )}{2} + 4 ln tan (\frac{x}{2}) + \frac{1}{2 tan ^{2} ( \frac{x}{2} )} + \frac{1}{2 tan ^{4} ( \frac{x}{2} )} - \frac{1}{6 tan ^{6} ( \frac{x}{2} )})

Vereinfache

\frac{1}{64} (\frac{tan ^{6} ( \frac{x}{2} )}{6} - \frac{tan ^{4} ( \frac{x}{2} )}{2} - \frac{tan ^{2} ( \frac{x}{2} )}{2} + 4 ln tan (\frac{x}{2}) + \frac{1}{2 tan ^{2} ( \frac{x}{2} )} + \frac{1}{2 tan ^{4} ( \frac{x}{2} )} - \frac{1}{6 tan ^{6} ( \frac{x}{2} )}) : \frac{1}{384} (tan^{6} (\frac{x}{2}) - 3 tan^{4} (\frac{x}{2}) - 3 tan^{2} (\frac{x}{2}) + 24 ln tan (\frac{x}{2}) + 3 cot^{2} (\frac{x}{2}) + 3 cot^{4} (\frac{x}{2}) - cot^{6} (\frac{x}{2}))

= \frac{1}{384} (tan^{6} (\frac{x}{2}) - 3 tan^{4} (\frac{x}{2}) - 3 tan^{2} (\frac{x}{2}) + 24 ln tan (\frac{x}{2}) + 3 cot^{2} (\frac{x}{2}) + 3 cot^{4} (\frac{x}{2}) - cot^{6} (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{384} (tan^{6} (\frac{x}{2}) - 3 tan^{4} (\frac{x}{2}) - 3 tan^{2} (\frac{x}{2}) + 24 ln tan (\frac{x}{2}) + 3 cot^{2} (\frac{x}{2}) + 3 cot^{4} (\frac{x}{2}) - cot^{6} (\frac{x}{2})) + C

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