integral of 3/2 x2^x

解

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

\frac{3}{ln ^{2} ( 2 )} \cdot 2^{x - 1} (ln (2) x - 1) + C

解答ステップ

\int \frac{3}{2} x \cdot 2^{x} d x

定数を除く:

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

= \frac{3}{2} \cdot \int x \cdot 2^{x} d x

部分積分を適用する

= \frac{3}{2} (\frac{2 ^{x} x}{ln ( 2 )} - \int \frac{2 ^{x}}{ln ( 2 )} d x)

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

= \frac{3}{2} (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )})

簡素化

\frac{3}{2} (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )}) : \frac{3}{ln ^{2} ( 2 )} \cdot 2^{x - 1} (ln (2) x - 1)

= \frac{3}{ln ^{2} ( 2 )} \cdot 2^{x - 1} (ln (2) x - 1)

定数を解答に追加する

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