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

Lösung

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int 2 \cdot (2 x + 3)^{3} d x

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\int x^{2} (x^{3} + 1)^{100} d x