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受欢迎的微积分 >

integral of sqrt((x^3-8)/(x^{11))}

解答

\int \frac{x ^{3} - 8}{x ^{11}} d x

解答

\frac{( x ^{3} - 8 ) ^{\frac{3}{2}}}{36 x ^{\frac{9}{2}}} + C

求解步骤

\int \frac{x ^{3} - 8}{x ^{11}} d x

化简

\frac{x ^{3} - 8}{x ^{11}} : \frac{x ^{3} - 8}{x ^{11}}

= \int \frac{x ^{3} - 8}{x ^{11}} d x

使用换元积分法

= \int \frac{u}{3 ( u + 8 ) ^{\frac{5}{2}}} d u

提出常数:

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

= \frac{1}{3} \cdot \int \frac{u}{( u + 8 ) ^{\frac{5}{2}}} d u

使用换元积分法

= \frac{1}{3} \cdot \int \frac{2 v ^{2}}{( v ^{2} + 8 ) ^{\frac{5}{2}}} d v

提出常数:

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

= \frac{1}{3} \cdot 2 \cdot \int \frac{v ^{2}}{( v ^{2} + 8 ) ^{\frac{5}{2}}} d v

使用三角换元法

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

提出常数:

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

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

使用三角恒等式改写

= \frac{1}{3} \cdot 2 \cdot \frac{1}{8} \cdot \int sin^{2} (w) cos (w) d w

使用换元积分法

= \frac{1}{3} \cdot 2 \cdot \frac{1}{8} \cdot \int t^{2} d t

使用幂法则

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

代回

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

化简

\frac{1}{3} \cdot 2 \cdot \frac{1}{8} \cdot \frac{sin ^{3} ( arctan ( \frac{1}{2 2} x ^{3} - 8 ) )}{3} : \frac{( x ^{3} - 8 ) ^{\frac{3}{2}}}{36 x ^{\frac{9}{2}}}

= \frac{( x ^{3} - 8 ) ^{\frac{3}{2}}}{36 x ^{\frac{9}{2}}}

解答补常数

= \frac{( x ^{3} - 8 ) ^{\frac{3}{2}}}{36 x ^{\frac{9}{2}}} + C

作图

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