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受欢迎的微积分 >

integral of (sqrt(x^2+9))/(x^6)

解答

\int \frac{x ^{2} + 9}{x ^{6}} d x

解答

\frac{1}{81} - \frac{( 9 + x ^{2} ) ^{\frac{5}{2}}}{5 x ^{5}} + \frac{( 9 + x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}} + C

求解步骤

\int \frac{x ^{2} + 9}{x ^{6}} d x

使用三角换元法

= \int \frac{sec ^{3} ( u )}{81 tan ^{6} ( u )} d u

提出常数:

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

= \frac{1}{81} \cdot \int \frac{sec ^{3} ( u )}{tan ^{6} ( u )} d u

使用基本三角恒等式:

\frac{1}{tan ( x )} = cot (x)

= \frac{1}{81} \cdot \int cot^{6} (u) sec^{3} (u) d u

使用三角恒等式改写

= \frac{1}{81} \cdot \int \frac{( 1 - sin ^{2} ( u ) ) cos ( u )}{sin ^{6} ( u )} d u

使用换元积分法

= \frac{1}{81} \cdot \int \frac{1 - v ^{2}}{v ^{6}} d v

乘开

\frac{1 - v ^{2}}{v ^{6}} : \frac{1}{v ^{6}} - \frac{1}{v ^{4}}

= \frac{1}{81} \cdot \int \frac{1}{v ^{6}} - \frac{1}{v ^{4}} 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

= \frac{1}{81} (\int \frac{1}{v ^{6}} d v - \int \frac{1}{v ^{4}} d v)

\int \frac{1}{v ^{6}} d v = - \frac{1}{5 v ^{5}}

\int \frac{1}{v ^{4}} d v = - \frac{1}{3 v ^{3}}

= \frac{1}{81} (- \frac{1}{5 v ^{5}} - (- \frac{1}{3 v ^{3}}))

代回

= \frac{1}{81} (- \frac{1}{5 sin ^{5} ( arctan ( \frac{1}{3} x ) )} - (- \frac{1}{3 sin ^{3} ( arctan ( \frac{1}{3} x ) )}))

化简

\frac{1}{81} (- \frac{1}{5 sin ^{5} ( arctan ( \frac{1}{3} x ) )} - (- \frac{1}{3 sin ^{3} ( arctan ( \frac{1}{3} x ) )})) : \frac{1}{81} - \frac{( 9 + x ^{2} ) ^{\frac{5}{2}}}{5 x ^{5}} + \frac{( 9 + x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}}

= \frac{1}{81} - \frac{( 9 + x ^{2} ) ^{\frac{5}{2}}}{5 x ^{5}} + \frac{( 9 + x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}}

解答补常数

= \frac{1}{81} - \frac{( 9 + x ^{2} ) ^{\frac{5}{2}}}{5 x ^{5}} + \frac{( 9 + x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}} + C

作图

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