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受欢迎的微积分 >

integral of x8^{x^2-8}

解答

\int x 8^{x^{2} - 8} d x

解答

\frac{8 ^{x^{2}}}{100663296 ln ( 2 )} + C

求解步骤

\int x \cdot 8^{x^{2} - 8} d x

x 8^{x^{2} - 8} = 8^{- 8} x 8^{x^{2}}

= \int 8^{- 8} x \cdot 8^{x^{2}} d x

提出常数:

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

= 8^{- 8} \cdot \int x \cdot 8^{x^{2}} d x

8^{- 8} = \frac{1}{16777216}

= \frac{1}{16777216} \cdot \int x \cdot 8^{x^{2}} d x

使用换元积分法

= \frac{1}{16777216} \cdot \int \frac{u}{3 ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) + l n ( 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}{16777216} \cdot \frac{1}{3 ln ( 2 )} \cdot \int \frac{u}{2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} d u

使用分布积分法

= \frac{1}{16777216} \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)

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

= \frac{1}{16777216} \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))

u = 8^{x^{2}}

代回

= \frac{1}{16777216} \cdot \frac{1}{3 ln ( 2 )} \frac{( 8 ^{x^{2}} ) ^{2}}{2 ^{\frac{l n ( 2 \cdot 8 ^{x^{2}} ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} \cdot 8^{x^{2}})

化简

\frac{1}{16777216} \cdot \frac{1}{3 ln ( 2 )} \frac{( 8 ^{x^{2}} ) ^{2}}{2 ^{\frac{l n ( 2 \cdot 8 ^{x^{2}} ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} \cdot 8^{x^{2}}) : \frac{8 ^{x^{2}}}{100663296 ln ( 2 )}

= \frac{8 ^{x^{2}}}{100663296 ln ( 2 )}

解答补常数

= \frac{8 ^{x^{2}}}{100663296 ln ( 2 )} + C

作图

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