integral of t^6(3+t^7)^6

解答

\int t^{6} (3 + t^{7})^{6} d t

\frac{729 t ^{7}}{7} + \frac{729 t ^{14}}{7} + \frac{405 t ^{21}}{7} + \frac{135 t ^{28}}{7} + \frac{27 t ^{35}}{7} + \frac{3 t ^{42}}{7} + \frac{t ^{49}}{49} + C

求解步骤

\int t^{6} (3 + t^{7})^{6} d t

乘开

t^{6} (3 + t^{7})^{6} : 729 t^{6} + 1458 t^{13} + 1215 t^{20} + 540 t^{27} + 135 t^{34} + 18 t^{41} + t^{48}

= \int 729 t^{6} + 1458 t^{13} + 1215 t^{20} + 540 t^{27} + 135 t^{34} + 18 t^{41} + t^{48} d t

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 729 t^{6} d t + \int 1458 t^{13} d t + \int 1215 t^{20} d t + \int 540 t^{27} d t + \int 135 t^{34} d t + \int 18 t^{41} d t + \int t^{48} d t

\int 729 t^{6} d t = \frac{729 t ^{7}}{7}

\int 1458 t^{13} d t = \frac{729 t ^{14}}{7}

\int 1215 t^{20} d t = \frac{405 t ^{21}}{7}

\int 540 t^{27} d t = \frac{135 t ^{28}}{7}

\int 135 t^{34} d t = \frac{27 t ^{35}}{7}

\int 18 t^{41} d t = \frac{3 t ^{42}}{7}

\int t^{48} d t = \frac{t ^{49}}{49}

= \frac{729 t ^{7}}{7} + \frac{729 t ^{14}}{7} + \frac{405 t ^{21}}{7} + \frac{135 t ^{28}}{7} + \frac{27 t ^{35}}{7} + \frac{3 t ^{42}}{7} + \frac{t ^{49}}{49}

解答补常数

= \frac{729 t ^{7}}{7} + \frac{729 t ^{14}}{7} + \frac{405 t ^{21}}{7} + \frac{135 t ^{28}}{7} + \frac{27 t ^{35}}{7} + \frac{3 t ^{42}}{7} + \frac{t ^{49}}{49} + C

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