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受欢迎的微积分 >

integral of sqrt((x^4-8)/(x^{14))}

解答

\int \frac{x ^{4} - 8}{x ^{14}} d x

解答

\frac{( x ^{4} - 8 ) ^{\frac{3}{2}}}{48 x ^{6}} + C

求解步骤

\int \frac{x ^{4} - 8}{x ^{14}} d x

使用换元积分法

= \int \frac{u ^{2} - 8}{2 u ^{4}} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{u ^{2} - 8}{u ^{4}} d u

使用三角换元法

= \frac{1}{2} \cdot \int \frac{tan ^{2} ( v )}{8 sec ^{3} ( v )} d v

提出常数:

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

= \frac{1}{2} \cdot \frac{1}{8} \cdot \int \frac{tan ^{2} ( v )}{sec ^{3} ( v )} d v

使用三角恒等式改写

= \frac{1}{2} \cdot \frac{1}{8} \cdot \int sin^{2} (v) cos (v) d v

使用换元积分法

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

使用幂法则

= \frac{1}{2} \cdot \frac{1}{8} \cdot \frac{w ^{3}}{3}

代回

= \frac{1}{2} \cdot \frac{1}{8} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{2 2} x ^{2} ) )}{3}

化简

\frac{1}{2} \cdot \frac{1}{8} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{2 2} x ^{2} ) )}{3} : \frac{( x ^{4} - 8 ) ^{\frac{3}{2}}}{48 x ^{6}}

= \frac{( x ^{4} - 8 ) ^{\frac{3}{2}}}{48 x ^{6}}

解答补常数

= \frac{( x ^{4} - 8 ) ^{\frac{3}{2}}}{48 x ^{6}} + C

作图

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