integral from x to y of 2^{ln(t)}

解

\int_{x}^{y} 2^{l n (t)} d t

\frac{( 2 e ) ^{l n (y)} - ( 2 e ) ^{l n (x)}}{ln ( 2 e )}

解答ステップ

\int_{x}^{y} 2^{l n (t)} d t

u置換積分法を適用する

= \int_{l n (x)}^{l n (y)} 2^{u} e^{u} d u

指数の規則を適用する:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

2^{u} e^{u} = (2 e)^{u}

= \int_{l n (x)}^{l n (y)} (2 e)^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= [\frac{( 2 e ) ^{u}}{ln ( 2 e )}]_{l n (x)}^{l n (y)}

限度を計算する:

\frac{( 2 e ) ^{l n (y)}}{ln ( 2 e )} - \frac{( 2 e ) ^{l n (x)}}{ln ( 2 e )}

= \frac{( 2 e ) ^{l n (y)}}{ln ( 2 e )} - \frac{( 2 e ) ^{l n (x)}}{ln ( 2 e )}

簡素化

= \frac{( 2 e ) ^{l n (y)} - ( 2 e ) ^{l n (x)}}{ln ( 2 e )}