integral of sin(ax)*e^{bx}

Lösung

\int sin (a x) \cdot e^{b x} d x

- \frac{a e ^{b x} cos ( a x )}{a ^{2} + b ^{2}} + \frac{b e ^{b x} sin ( a x )}{a ^{2} + b ^{2}} + C

Schritte zur Lösung

\int sin (a x) e^{b x} d x

Wende die partielle Integration an

= - e^{b x} \frac{1}{a} cos (a x) - \int - b e^{b x} \frac{1}{a} cos (a x) d x

Entferne die Konstante:

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

= - e^{b x} \frac{1}{a} cos (a x) - (- b \frac{1}{a} \cdot \int e^{b x} cos (a x) d x)

Vereinfache

- b \frac{1}{a} : - \frac{b}{a}

= - e^{b x} \frac{1}{a} cos (a x) - (- \frac{b}{a} \cdot \int e^{b x} cos (a x) d x)

Wende die partielle Integration an

= - e^{b x} \frac{1}{a} cos (a x) - (- \frac{b}{a} (e^{b x} \frac{1}{a} sin (a x) - \int e^{b x} b \frac{1}{a} sin (a x) d x))

Entferne die Konstante:

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

= - e^{b x} \frac{1}{a} cos (a x) - (- \frac{b}{a} (e^{b x} \frac{1}{a} sin (a x) - b \frac{1}{a} \cdot \int e^{b x} sin (a x) d x))

Vereinfache

b \frac{1}{a} : \frac{b}{a}

= - e^{b x} \frac{1}{a} cos (a x) - (- \frac{b}{a} (e^{b x} \frac{1}{a} sin (a x) - \frac{b}{a} \cdot \int e^{b x} sin (a x) d x))

Deshalb

\int e^{b x} sin (a x) d x = - e^{b x} \frac{1}{a} cos (a x) - (- \frac{b}{a} (e^{b x} \frac{1}{a} sin (a x) - \frac{b}{a} \cdot \int e^{b x} sin (a x) d x))

Isoliere

\int e^{b x} sin (a x) d x

= - \frac{a e ^{b x} cos ( a x )}{a ^{2} + b ^{2}} + \frac{b e ^{b x} sin ( a x )}{a ^{2} + b ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{a e ^{b x} cos ( a x )}{a ^{2} + b ^{2}} + \frac{b e ^{b x} sin ( a x )}{a ^{2} + b ^{2}} + C

\int (cos (a x) + sin (a x))^{2} d x

\int \frac{x ^{2} + 2 x + 1}{x ^{3} - 16 x} d x

\int \frac{4 x}{3 + 2 x ^{2}} d x

\int (x e^{- 3 x}) d x

\int cot (29 x) d x