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受欢迎的微积分 >

integral of x^3(x^2+1)^{10}

解答

\int x^{3} (x^{2} + 1)^{10} d x

解答

\frac{1}{264} (11 x^{24} + 120 x^{22} + 594 x^{20} + 1760 x^{18} + 3465 x^{16} + 4752 x^{14} + 4620 x^{12} + 3168 x^{10} + 1485 x^{8} + 440 x^{6} + 66 x^{4}) + C

求解步骤

\int x^{3} (x^{2} + 1)^{10} d x

使用换元积分法

= \int \frac{u ^{\frac{1}{3}} ( u ^{\frac{2}{3}} + 1 ) ^{10}}{3} 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 u^{\frac{1}{3}} (u^{\frac{2}{3}} + 1)^{10} d u

乘开

u^{\frac{1}{3}} (u^{\frac{2}{3}} + 1)^{10} : u^{7} + 10 u^{\frac{19}{3}} + 45 u^{\frac{17}{3}} + 120 u^{5} + 210 u^{\frac{13}{3}} + 252 u^{\frac{11}{3}} + 210 u^{3} + 120 u^{\frac{7}{3}} + 45 u^{\frac{5}{3}} + 10 u + u^{\frac{1}{3}}

= \frac{1}{3} \cdot \int u^{7} + 10 u^{\frac{19}{3}} + 45 u^{\frac{17}{3}} + 120 u^{5} + 210 u^{\frac{13}{3}} + 252 u^{\frac{11}{3}} + 210 u^{3} + 120 u^{\frac{7}{3}} + 45 u^{\frac{5}{3}} + 10 u + u^{\frac{1}{3}} d u

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}{3} (\int u^{7} d u + \int 10 u^{\frac{19}{3}} d u + \int 45 u^{\frac{17}{3}} d u + \int 120 u^{5} d u + \int 210 u^{\frac{13}{3}} d u + \int 252 u^{\frac{11}{3}} d u + \int 210 u^{3} d u + \int 120 u^{\frac{7}{3}} d u + \int 45 u^{\frac{5}{3}} d u + \int 10 u d u + \int u^{\frac{1}{3}} d u)

\int u^{7} d u = \frac{u ^{8}}{8}

\int 10 u^{\frac{19}{3}} d u = \frac{15}{11} u^{\frac{22}{3}}

\int 45 u^{\frac{17}{3}} d u = \frac{27}{4} u^{\frac{20}{3}}

\int 120 u^{5} d u = 20 u^{6}

\int 210 u^{\frac{13}{3}} d u = \frac{315}{8} u^{\frac{16}{3}}

\int 252 u^{\frac{11}{3}} d u = 54 u^{\frac{14}{3}}

\int 210 u^{3} d u = \frac{105 u ^{4}}{2}

\int 120 u^{\frac{7}{3}} d u = 36 u^{\frac{10}{3}}

\int 45 u^{\frac{5}{3}} d u = \frac{135}{8} u^{\frac{8}{3}}

\int 10 u d u = 5 u^{2}

\int u^{\frac{1}{3}} d u = \frac{3}{4} u^{\frac{4}{3}}

= \frac{1}{3} (\frac{u ^{8}}{8} + \frac{15}{11} u^{\frac{22}{3}} + \frac{27}{4} u^{\frac{20}{3}} + 20 u^{6} + \frac{315}{8} u^{\frac{16}{3}} + 54 u^{\frac{14}{3}} + \frac{105 u ^{4}}{2} + 36 u^{\frac{10}{3}} + \frac{135}{8} u^{\frac{8}{3}} + 5 u^{2} + \frac{3}{4} u^{\frac{4}{3}})

u = x^{3}

代回

= \frac{1}{3} (\frac{( x ^{3} ) ^{8}}{8} + \frac{15}{11} (x^{3})^{\frac{22}{3}} + \frac{27}{4} (x^{3})^{\frac{20}{3}} + 20 (x^{3})^{6} + \frac{315}{8} (x^{3})^{\frac{16}{3}} + 54 (x^{3})^{\frac{14}{3}} + \frac{105 ( x ^{3} ) ^{4}}{2} + 36 (x^{3})^{\frac{10}{3}} + \frac{135}{8} (x^{3})^{\frac{8}{3}} + 5 (x^{3})^{2} + \frac{3}{4} (x^{3})^{\frac{4}{3}})

化简

\frac{1}{3} (\frac{( x ^{3} ) ^{8}}{8} + \frac{15}{11} (x^{3})^{\frac{22}{3}} + \frac{27}{4} (x^{3})^{\frac{20}{3}} + 20 (x^{3})^{6} + \frac{315}{8} (x^{3})^{\frac{16}{3}} + 54 (x^{3})^{\frac{14}{3}} + \frac{105 ( x ^{3} ) ^{4}}{2} + 36 (x^{3})^{\frac{10}{3}} + \frac{135}{8} (x^{3})^{\frac{8}{3}} + 5 (x^{3})^{2} + \frac{3}{4} (x^{3})^{\frac{4}{3}}) : \frac{1}{264} (11 x^{24} + 120 x^{22} + 594 x^{20} + 1760 x^{18} + 3465 x^{16} + 4752 x^{14} + 4620 x^{12} + 3168 x^{10} + 1485 x^{8} + 440 x^{6} + 66 x^{4})

= \frac{1}{264} (11 x^{24} + 120 x^{22} + 594 x^{20} + 1760 x^{18} + 3465 x^{16} + 4752 x^{14} + 4620 x^{12} + 3168 x^{10} + 1485 x^{8} + 440 x^{6} + 66 x^{4})

解答补常数

= \frac{1}{264} (11 x^{24} + 120 x^{22} + 594 x^{20} + 1760 x^{18} + 3465 x^{16} + 4752 x^{14} + 4620 x^{12} + 3168 x^{10} + 1485 x^{8} + 440 x^{6} + 66 x^{4}) + C

作图

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