integral of 3^xe^xsin(x)

解

\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

解答ステップ

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

部分積分を適用する

= \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})

簡素化

= \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}

定数を解答に追加する

= \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