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受欢迎的微积分 >

integral from 1 to sqrt(2 of)x4^{x^2}

解答

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

解答

\frac{3}{ln ( 2 )}

+1

十进制

4.32808 \dots

求解步骤

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

使用换元积分法

= \int_{4}^{16} \frac{u}{ln ( 2 ) \cdot 4 ^{\frac{l n ( u ) + 2 l n ( 2 )}{2 l n ( 2 )}}} d u

提出常数:

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

= \frac{1}{ln ( 2 )} \cdot \int_{4}^{16} \frac{u}{4 ^{\frac{l n ( u ) + 2 l n ( 2 )}{2 l n ( 2 )}}} d u

使用分布积分法

= \frac{1}{ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 4 u ) + l n ( 2 )}{l n ( 2 )}}} - \int - \frac{1}{8} d u]_{4}^{16}

\int - \frac{1}{8} d u = - \frac{1}{8} u

= \frac{1}{ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 4 u ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{8} u)]_{4}^{16}

化简

= \frac{1}{ln ( 2 )} [\frac{u ^{2}}{2 ^{\frac{l n ( 4 u ) + l n ( 2 )}{l n ( 2 )}}} + \frac{1}{8} u]_{4}^{16}

带入上下限计算后得:

3

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

化简

= \frac{3}{ln ( 2 )}

作图

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