integral of (csc(x)+cot(x))^2

Lösung

\int (csc (x) + cot (x))^{2} d x

2 (- cot (\frac{x}{2}) - arctan (tan (\frac{x}{2}))) + C

Schritte zur Lösung

\int (csc (x) + cot (x))^{2} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= 2 \cdot \int \frac{1}{u ^{2} ( 1 + u ^{2} )} d u

Ermittle den Partialbruch von

\frac{1}{u ^{2} ( 1 + u ^{2} )} : \frac{1}{u ^{2}} - \frac{1}{u ^{2} + 1}

= 2 \cdot \int \frac{1}{u ^{2}} - \frac{1}{u ^{2} + 1} 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

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

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

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 2 (- \frac{1}{u} - arctan (u))

Setze in

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

ein

= 2 (- \frac{1}{tan ( \frac{x}{2} )} - arctan (tan (\frac{x}{2})))

Verwende die grundlegende trigonometrische Identität:

\frac{1}{tan ( x )} = cot (x)

= 2 (- cot (\frac{x}{2}) - arctan (tan (\frac{x}{2})))

Füge eine Konstante zur Lösung hinzu

= 2 (- cot (\frac{x}{2}) - arctan (tan (\frac{x}{2}))) + C

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