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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int ln (\frac{x}{1 - x}) d x

\int sin^{4} (\frac{x}{2}) cos (\frac{x}{2}) d x

\int (tan (x) sec (x)) d x

\int \frac{x ^{2} + cos ( x )}{x ^{3} + 3 sin ( x )} d x

\int x ln (4) d x