integral of sin(x)cos(x)e^{sin^2(x)+pi}

Lösung

\int sin (x) cos (x) e^{s i n^{2} (x) + π} d x

\frac{1}{2} e^{π + s i n^{2} (x)} + C

Schritte zur Lösung

\int sin (x) cos (x) e^{s i n^{2} (x) + π} d x

sin (x) cos (x) e^{s i n^{2} (x) + π} = e^{π} sin (x) cos (x) e^{s i n^{2} (x)}

= \int e^{π} sin (x) cos (x) e^{s i n^{2} (x)} d x

Entferne die Konstante:

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

= e^{π} \cdot \int sin (x) cos (x) e^{s i n^{2} (x)} d x

Wende U-Substitution an

= e^{π} \cdot \int e^{u^{2}} u d u

Wende U-Substitution an

= e^{π} \cdot \int \frac{e ^{v}}{2} d v

Entferne die Konstante:

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

= e^{π} \frac{1}{2} \cdot \int e^{v} d v

Nutze das gemeinsame Integral :

\int e^{v} d v = e^{v}

= e^{π} \frac{1}{2} e^{v}

Ersetze zurück

= e^{π} \frac{1}{2} e^{s i n^{2} (x)}

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

e^{π} e^{s i n^{2} (x)} = e^{π + s i n^{2} (x)}

= \frac{1}{2} e^{π + s i n^{2} (x)}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} e^{π + s i n^{2} (x)} + C