integral of cos(x/2)*cos(nx)

Lösung

\int cos (\frac{x}{2}) \cdot cos (n x) d x

\frac{1}{n} sin (n x) cos (\frac{x}{2}) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2})) + C

Schritte zur Lösung

\int cos (\frac{x}{2}) cos (n x) d x

Wende die partielle Integration an

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

\int - \frac{1}{2 n} sin (n x) sin (\frac{x}{2}) d x = - \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2}))

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) - (- \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2})))

Vereinfache

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{n} sin (n x) cos (\frac{x}{2}) + \frac{1}{4 n} (- \frac{2}{2 n + 1} sin (n x + \frac{x}{2}) + \frac{2}{2 n - 1} sin (n x - \frac{x}{2})) + C

\int t^{2} e^{t} d t

n = 1 \sum \infty \frac{n - 1}{n}

\int 12 x^{2} + 6 x - 4 d x

\frac{\partial}{\partial x} (y cos (x y) - y sin (x y))

h \to 0 lim (\frac{( sin ( \frac{π}{6} + h ) - \frac{1}{2} )}{h})