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人気のある微分積分 >

integral from 1 to 2 of 27x^24^{x^3}

解

\int_{1}^{2} 27 x^{2} 4^{x^{3}} d x

解

\frac{294894}{ln ( 2 )}

+1

十進法表記

425442.11138 \dots

解答ステップ

\int_{1}^{2} 27 x^{2} \cdot 4^{x^{3}} d x

定数を除く:

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

= 27 \cdot \int_{1}^{2} x^{2} \cdot 4^{x^{3}} d x

u置換積分法を適用する

= 27 \cdot \int_{4}^{65536} \frac{u}{3 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

= 27 \cdot \frac{1}{3 ln ( 2 )} \cdot \int_{4}^{65536} \frac{u}{2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} d u

部分積分を適用する

= 27 \cdot \frac{1}{3 ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} - \int - \frac{1}{4} d u]_{4}^{65536}

\int - \frac{1}{4} d u = - \frac{1}{4} u

= 27 \cdot \frac{1}{3 ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} u)]_{4}^{65536}

簡素化

27 \cdot \frac{1}{3 ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} u)]_{4}^{65536} : \frac{9}{ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} + \frac{1}{4} u]_{4}^{65536}

= \frac{9}{ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} + \frac{1}{4} u]_{4}^{65536}

限度を計算する:

32766

= \frac{9}{ln ( 2 )} \cdot 32766

簡素化

= \frac{294894}{ln ( 2 )}

グラフ

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