integral of xe^{x^2}cos(x^2)

Lösung

\int x e^{x^{2}} cos (x^{2}) d x

\frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2})) + C

Schritte zur Lösung

\int x e^{x^{2}} cos (x^{2}) d x

Wende U-Substitution an

= \int \frac{e ^{u} cos ( u )}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int e^{u} cos (u) d u

Wende die partielle Integration an

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

Wende die partielle Integration an

= \frac{1}{2} (e^{u} sin (u) - (- e^{u} cos (u) - \int - e^{u} cos (u) d u))

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} (e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u)))

Deshalb

\frac{1}{2} \cdot \int e^{u} cos (u) d u = \frac{1}{2} (e^{u} sin (u) - (- e^{u} cos (u) - (- \int e^{u} cos (u) d u)))

Isoliere

\int e^{u} cos (u) d u

= \frac{1}{2} (\frac{e ^{u} sin ( u )}{2} + \frac{e ^{u} cos ( u )}{2})

Setze in

u = x^{2}

ein

= \frac{1}{2} (\frac{e ^{x^{2}} sin ( x ^{2} )}{2} + \frac{e ^{x^{2}} cos ( x ^{2} )}{2})

Vereinfache

\frac{1}{2} (\frac{e ^{x^{2}} sin ( x ^{2} )}{2} + \frac{e ^{x^{2}} cos ( x ^{2} )}{2}) : \frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2}))

= \frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2})) + C

\frac{d}{d x} (\frac{1 - 2 x}{3 x ^{2}})

\frac{d}{d x} ((\frac{x + 5}{x ^{2} + 2})^{2})

\frac{d ^{2}}{d x ^{2}} (e^{\frac{6}{x}})

x \to \infty lim (- 3 x^{3} + x)

a re a y = 0, y = 3 (x^{3} - x)