integral of 3^xe^xsin(x)

Lösung

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

\frac{sin ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} - \frac{1}{ln ( 3 e ) + ln ^{3} ( 3 e )} sin (x) (3 e)^{x} - \frac{1}{1 + ln ^{2} ( 3 e )} cos (x) (3 e)^{x} + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{sin ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} - \int \frac{cos ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} d x

\int \frac{cos ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} d x = \frac{1}{ln ( 3 e ) + ln ^{3} ( 3 e )} sin (x) (3 e)^{x} + \frac{1}{1 + ln ^{2} ( 3 e )} cos (x) (3 e)^{x}

= \frac{sin ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} - (\frac{1}{ln ( 3 e ) + ln ^{3} ( 3 e )} sin (x) (3 e)^{x} + \frac{1}{1 + ln ^{2} ( 3 e )} cos (x) (3 e)^{x})

Vereinfache

= \frac{sin ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} - \frac{1}{ln ( 3 e ) + ln ^{3} ( 3 e )} sin (x) (3 e)^{x} - \frac{1}{1 + ln ^{2} ( 3 e )} cos (x) (3 e)^{x}

Füge eine Konstante zur Lösung hinzu

= \frac{sin ( x ) ( 3 e ) ^{x}}{ln ( 3 e )} - \frac{1}{ln ( 3 e ) + ln ^{3} ( 3 e )} sin (x) (3 e)^{x} - \frac{1}{1 + ln ^{2} ( 3 e )} cos (x) (3 e)^{x} + C

\int \frac{5 + x ^{2}}{x} d x

\int sin (3 x) - tan (2 x) + \frac{5}{3 x + 2} d x

\int 3 - 0.01 x + 0.00006 x^{2} d x

\int cos^{3} (12 x) d x

\int [ln (q)] d q