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受欢迎的微积分 >

integral of 1/(x^2sqrt(2-3x))

解答

\int \frac{1}{x ^{2} 2 - 3 x} d x

解答

- 48 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )}) + C

求解步骤

\int \frac{1}{x ^{2} 2 - 3 x} d x

使用换元积分法

= \int - \frac{6}{( u ^{2} - 2 ) ^{2}} d u

提出常数:

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

= - 6 \cdot \int \frac{1}{( u ^{2} - 2 ) ^{2}} d u

使用换元积分法

= - 6 \cdot \int \frac{8}{( v ^{2} - 8 ) ^{2}} d v

提出常数:

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

= - 6 \cdot 8 \cdot \int \frac{1}{( v ^{2} - 8 ) ^{2}} d v

将

\frac{1}{( v ^{2} - 8 ) ^{2}}

用部份分式展开:

\frac{1}{64 2 ( v + 2 2 )} + \frac{1}{32 ( v + 2 2 ) ^{2}} - \frac{1}{64 2 ( v - 2 2 )} + \frac{1}{32 ( v - 2 2 ) ^{2}}

= - 6 \cdot 8 \cdot \int \frac{1}{64 2 ( v + 2 2 )} + \frac{1}{32 ( v + 2 2 ) ^{2}} - \frac{1}{64 2 ( v - 2 2 )} + \frac{1}{32 ( v - 2 2 ) ^{2}} d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 6 \cdot 8 (\int \frac{1}{64 2 ( v + 2 2 )} d v + \int \frac{1}{32 ( v + 2 2 ) ^{2}} d v - \int \frac{1}{64 2 ( v - 2 2 )} d v + \int \frac{1}{32 ( v - 2 2 ) ^{2}} d v)

\int \frac{1}{64 2 ( v + 2 2 )} d v = \frac{1}{64 2} ln v + 22

\int \frac{1}{32 ( v + 2 2 ) ^{2}} d v = - \frac{1}{32 ( v + 2 2 )}

\int \frac{1}{64 2 ( v - 2 2 )} d v = \frac{1}{64 2} ln v - 22

\int \frac{1}{32 ( v - 2 2 ) ^{2}} d v = - \frac{1}{32 ( v - 2 2 )}

= - 6 \cdot 8 (\frac{1}{64 2} ln v + 22 - \frac{1}{32 ( v + 2 2 )} - \frac{1}{64 2} ln v - 22 - \frac{1}{32 ( v - 2 2 )})

代回

= - 6 \cdot 8 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )})

化简

= - 48 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )})

解答补常数

= - 48 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )}) + C

作图

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