integral of e^{2θ}sin(3θ)

Lösung

\int e^{2 θ} sin (3 θ) d θ

- \frac{3 e ^{2 θ} cos ( 3 θ )}{13} + \frac{2 e ^{2 θ} sin ( 3 θ )}{13} + C

Schritte zur Lösung

\int e^{2 θ} sin (3 θ) d θ

Wende die partielle Integration an

= - \frac{1}{3} e^{2 θ} cos (3 θ) - \int - \frac{2}{3} e^{2 θ} cos (3 θ) d θ

Entferne die Konstante:

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

= - \frac{1}{3} e^{2 θ} cos (3 θ) - (- \frac{2}{3} \cdot \int e^{2 θ} cos (3 θ) d θ)

Wende die partielle Integration an

= - \frac{1}{3} e^{2 θ} cos (3 θ) - (- \frac{2}{3} (\frac{1}{3} e^{2 θ} sin (3 θ) - \int \frac{2}{3} e^{2 θ} sin (3 θ) d θ))

Entferne die Konstante:

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

= - \frac{1}{3} e^{2 θ} cos (3 θ) - (- \frac{2}{3} (\frac{1}{3} e^{2 θ} sin (3 θ) - \frac{2}{3} \cdot \int e^{2 θ} sin (3 θ) d θ))

Deshalb

\int e^{2 θ} sin (3 θ) d θ = - \frac{1}{3} e^{2 θ} cos (3 θ) - (- \frac{2}{3} (\frac{1}{3} e^{2 θ} sin (3 θ) - \frac{2}{3} \cdot \int e^{2 θ} sin (3 θ) d θ))

Isoliere

\int e^{2 θ} sin (3 θ) d θ

= - \frac{3 e ^{2 θ} cos ( 3 θ )}{13} + \frac{2 e ^{2 θ} sin ( 3 θ )}{13}

Füge eine Konstante zur Lösung hinzu

= - \frac{3 e ^{2 θ} cos ( 3 θ )}{13} + \frac{2 e ^{2 θ} sin ( 3 θ )}{13} + C

(ln (f (x)))^{'}

in v erse l a pl a ce 10 s

(3^{- x^{2}})^{^{'}}

d er i v a t i v e 1 - \frac{36}{x ^{2}}

\frac{d x}{d t} = a x + b