integral of xe^{inx}

Lösung

\int x e^{in x} d x

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

Schritte zur Lösung

\int x e^{in x} d x

Vereinfache

x e^{in x} : x cos (n x) + i x sin (n x)

= \int x cos (n x) + i x sin (n 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 x cos (n x) d x + \int i x sin (n x) d x

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{100 x}{x ^{2} + 2} d x

\int \frac{tan ^{2} ( x )}{csc ( x )} d x

\int ((sin (x))^{2}) d x

\int x - 9 d x

\int 8 cos (x) sin (x) d x