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受欢迎的微积分 >

integral of 1/((x^2-6x+10)^{5/2)}

解答

\int \frac{1}{( x ^{2} - 6 x + 10 ) ^{\frac{5}{2}}} d x

解答

\frac{x - 3}{( x ^{2} - 6 x + 10 ) ^{\frac{1}{2}}} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}} + C

求解步骤

\int \frac{1}{( x ^{2} - 6 x + 10 ) ^{\frac{5}{2}}} d x

对

x^{2} - 6 x + 10

配方:

(x - 3)^{2} + 1

= \int \frac{1}{( ( x - 3 ) ^{2} + 1 ) ^{\frac{5}{2}}} d x

使用换元积分法

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

使用三角换元法

= \int cos^{3} (v) d v

使用三角恒等式改写

= \int (1 - sin^{2} (v)) cos (v) d v

使用换元积分法

= \int 1 - w^{2} d w

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

= \int 1 d w - \int w^{2} d w

\int 1 d w = w

\int w^{2} d w = \frac{w ^{3}}{3}

= w - \frac{w ^{3}}{3}

代回

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

化简

sin (arctan (x - 3)) - \frac{sin ^{3} ( arctan ( x - 3 ) )}{3} : \frac{x - 3}{( x ^{2} - 6 x + 10 ) ^{\frac{1}{2}}} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}}

= \frac{x - 3}{( x ^{2} - 6 x + 10 ) ^{\frac{1}{2}}} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}}

解答补常数

= \frac{x - 3}{( x ^{2} - 6 x + 10 ) ^{\frac{1}{2}}} - \frac{( x - 3 ) ^{3}}{3 ( x ^{2} - 6 x + 10 ) ^{\frac{3}{2}}} + C

作图

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