integral of e^{sin(x)+1}*cos(x)

Lösung

\int e^{s i n (x) + 1} \cdot cos (x) d x

e^{1 + s i n (x)} + C

Schritte zur Lösung

\int e^{s i n (x) + 1} cos (x) d x

e^{s i n (x) + 1} cos (x) = e^{1} e^{s i n (x)} cos (x)

= \int e^{1} e^{s i n (x)} cos (x) d x

Entferne die Konstante:

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

= e^{1} \cdot \int e^{s i n (x)} cos (x) d x

Vereinfache

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

Wende U-Substitution an

= e \cdot \int e^{u} d u

Nutze das gemeinsame Integral :

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

= e e^{u}

Setze in

u = sin (x)

ein

= e e^{s i n (x)}

Wende Exponentenregel an:

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

e e^{s i n (x)} = e^{1 + s i n (x)}

= e^{1 + s i n (x)}

Füge eine Konstante zur Lösung hinzu

= e^{1 + s i n (x)} + C

\int \frac{1}{x ( ln ( x ) ) ^{\frac{1}{2}}} d x

\int \frac{3 x + 1}{x ^{2} - x - 6} d x

\int e^{- \frac{3}{5} x} d x

\int 12 x^{5} y^{2} d x

\int \frac{1}{x 1 - 2 x ^{2}} d x