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受欢迎的微积分 >

integral of x^3(3x^4+2)^9

解答

\int x^{3} (3 x^{4} + 2)^{9} d x

解答

\frac{1}{40} x^{4} (19683 x^{36} + 131220 x^{32} + 393660 x^{28} + 699840 x^{24} + 816480 x^{20} + 653184 x^{16} + 362880 x^{12} + 138240 x^{8} + 34560 x^{4} + 5120) + C

求解步骤

\int x^{3} (3 x^{4} + 2)^{9} d x

使用换元积分法

= \int \frac{u ^{\frac{1}{3}} ( 3 u ^{\frac{4}{3}} + 2 ) ^{9}}{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}} (3 u^{\frac{4}{3}} + 2)^{9} d u

乘开

u^{\frac{1}{3}} (3 u^{\frac{4}{3}} + 2)^{9} : 19683 u^{\frac{37}{3}} + 118098 u^{11} + 314928 u^{\frac{29}{3}} + 489888 u^{\frac{25}{3}} + 489888 u^{7} + 326592 u^{\frac{17}{3}} + 145152 u^{\frac{13}{3}} + 41472 u^{3} + 6912 u^{\frac{5}{3}} + 512 u^{\frac{1}{3}}

= \frac{1}{3} \cdot \int 19683 u^{\frac{37}{3}} + 118098 u^{11} + 314928 u^{\frac{29}{3}} + 489888 u^{\frac{25}{3}} + 489888 u^{7} + 326592 u^{\frac{17}{3}} + 145152 u^{\frac{13}{3}} + 41472 u^{3} + 6912 u^{\frac{5}{3}} + 512 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 19683 u^{\frac{37}{3}} d u + \int 118098 u^{11} d u + \int 314928 u^{\frac{29}{3}} d u + \int 489888 u^{\frac{25}{3}} d u + \int 489888 u^{7} d u + \int 326592 u^{\frac{17}{3}} d u + \int 145152 u^{\frac{13}{3}} d u + \int 41472 u^{3} d u + \int 6912 u^{\frac{5}{3}} d u + \int 512 u^{\frac{1}{3}} d u)

\int 19683 u^{\frac{37}{3}} d u = \frac{59049}{40} u^{\frac{40}{3}}

\int 118098 u^{11} d u = \frac{19683 u ^{12}}{2}

\int 314928 u^{\frac{29}{3}} d u = \frac{59049}{2} u^{\frac{32}{3}}

\int 489888 u^{\frac{25}{3}} d u = 52488 u^{\frac{28}{3}}

\int 489888 u^{7} d u = 61236 u^{8}

\int 326592 u^{\frac{17}{3}} d u = \frac{244944}{5} u^{\frac{20}{3}}

\int 145152 u^{\frac{13}{3}} d u = 27216 u^{\frac{16}{3}}

\int 41472 u^{3} d u = 10368 u^{4}

\int 6912 u^{\frac{5}{3}} d u = 2592 u^{\frac{8}{3}}

\int 512 u^{\frac{1}{3}} d u = 384 u^{\frac{4}{3}}

= \frac{1}{3} (\frac{59049}{40} u^{\frac{40}{3}} + \frac{19683 u ^{12}}{2} + \frac{59049}{2} u^{\frac{32}{3}} + 52488 u^{\frac{28}{3}} + 61236 u^{8} + \frac{244944}{5} u^{\frac{20}{3}} + 27216 u^{\frac{16}{3}} + 10368 u^{4} + 2592 u^{\frac{8}{3}} + 384 u^{\frac{4}{3}})

u = x^{3}

代回

= \frac{1}{3} (\frac{59049}{40} (x^{3})^{\frac{40}{3}} + \frac{19683 ( x ^{3} ) ^{12}}{2} + \frac{59049}{2} (x^{3})^{\frac{32}{3}} + 52488 (x^{3})^{\frac{28}{3}} + 61236 (x^{3})^{8} + \frac{244944}{5} (x^{3})^{\frac{20}{3}} + 27216 (x^{3})^{\frac{16}{3}} + 10368 (x^{3})^{4} + 2592 (x^{3})^{\frac{8}{3}} + 384 (x^{3})^{\frac{4}{3}})

化简

\frac{1}{3} (\frac{59049}{40} (x^{3})^{\frac{40}{3}} + \frac{19683 ( x ^{3} ) ^{12}}{2} + \frac{59049}{2} (x^{3})^{\frac{32}{3}} + 52488 (x^{3})^{\frac{28}{3}} + 61236 (x^{3})^{8} + \frac{244944}{5} (x^{3})^{\frac{20}{3}} + 27216 (x^{3})^{\frac{16}{3}} + 10368 (x^{3})^{4} + 2592 (x^{3})^{\frac{8}{3}} + 384 (x^{3})^{\frac{4}{3}}) : \frac{1}{40} x^{4} (19683 x^{36} + 131220 x^{32} + 393660 x^{28} + 699840 x^{24} + 816480 x^{20} + 653184 x^{16} + 362880 x^{12} + 138240 x^{8} + 34560 x^{4} + 5120)

= \frac{1}{40} x^{4} (19683 x^{36} + 131220 x^{32} + 393660 x^{28} + 699840 x^{24} + 816480 x^{20} + 653184 x^{16} + 362880 x^{12} + 138240 x^{8} + 34560 x^{4} + 5120)

解答补常数

= \frac{1}{40} x^{4} (19683 x^{36} + 131220 x^{32} + 393660 x^{28} + 699840 x^{24} + 816480 x^{20} + 653184 x^{16} + 362880 x^{12} + 138240 x^{8} + 34560 x^{4} + 5120) + C

作图

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