integral of x-(1/2)^x

解

\int x - (\frac{1}{2})^{x} d x

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

解答ステップ

\int x - (\frac{1}{2})^{x} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x d x - \int (\frac{1}{2})^{x} d x

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

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

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

簡素化

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

定数を解答に追加する

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