integral of sin(pix)sin(npix)

Lösung

\int sin (π x) sin (nπ x) d x

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

Schritte zur Lösung

\int sin (π x) sin (nπ x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (x^{3} \cdot ln (x))

d er i v a t i v e y = y

roo t s \frac{k x}{a ^{2} + x ^{2}}

t an g e n t f (x) = x (x - 2), at x = 1

\int_{0}^{2} x^{3} e^{x} d x