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人気のある微分積分 >

integral of (2^{sqrt(x)})/(sqrt(x))

解

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

解

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

解答ステップ

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

u置換積分法を適用する

= \int \frac{u}{ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) - l n ( 2 )}{l n ( 2 )}}} d u

定数を除く:

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

= \frac{1}{ln ( 2 )} \cdot \int \frac{u}{2 ^{\frac{l n ( u ) - l n ( 2 )}{l n ( 2 )}}} d u

部分積分を適用する

= \frac{1}{ln ( 2 )} (\frac{u ^{2}}{2 ^{\frac{l n ( u )}{l n ( 2 )}}} - \int - 1 d u)

\int - 1 d u = - u

= \frac{1}{ln ( 2 )} (\frac{u ^{2}}{2 ^{\frac{l n ( u )}{l n ( 2 )}}} - (- u))

代用を戻す

u = 2^{x}

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of 1-6x

\int 1 - 6 x d x

integral from 0 to pi/4 of cot(x)

\int_{0}^{\frac{π}{4}} cot (x) d x

tangent f(x)=5x-4x^2,(-1,-9)

t an g e n t f (x) = 5 x - 4 x^{2}, (- 1, - 9)

sum from n=0 to infinity of 5/(2+3^n)

n = 0 \sum \infty \frac{5}{2 + 3 ^{n}}

integral of (csc(x)cot(x))/(2-csc(x))

\int \frac{csc ( x ) cot ( x )}{2 - csc ( x )} d x