integral of (4x^2+8x+9)^6(x+1)

解答

\int (4 x^{2} + 8 x + 9)^{6} (x + 1) d x

\frac{2048 x ^{14}}{7} + 4096 x^{13} + 29184 x^{12} + 137216 x^{11} + 471424 x^{10} + 1244928 x^{9} + 2597792 x^{8} + \frac{30405376 x ^{7}}{7} + 5845032 x^{6} + 6302448 x^{5} + 5369814 x^{4} + 3516696 x^{3} + \frac{3365793 x ^{2}}{2} + 531441 x + C

求解步骤

\int (4 x^{2} + 8 x + 9)^{6} (x + 1) d x

乘开

(4 x^{2} + 8 x + 9)^{6} (x + 1) : 4096 x^{13} + 53248 x^{12} + 350208 x^{11} + 1509376 x^{10} + 4714240 x^{9} + 11204352 x^{8} + 20782336 x^{7} + 30405376 x^{6} + 35070192 x^{5} + 31512240 x^{4} + 21479256 x^{3} + 10550088 x^{2} + 3365793 x + 531441

= \int 4096 x^{13} + 53248 x^{12} + 350208 x^{11} + 1509376 x^{10} + 4714240 x^{9} + 11204352 x^{8} + 20782336 x^{7} + 30405376 x^{6} + 35070192 x^{5} + 31512240 x^{4} + 21479256 x^{3} + 10550088 x^{2} + 3365793 x + 531441 d x

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 4096 x^{13} d x + \int 53248 x^{12} d x + \int 350208 x^{11} d x + \int 1509376 x^{10} d x + \int 4714240 x^{9} d x + \int 11204352 x^{8} d x + \int 20782336 x^{7} d x + \int 30405376 x^{6} d x + \int 35070192 x^{5} d x + \int 31512240 x^{4} d x + \int 21479256 x^{3} d x + \int 10550088 x^{2} d x + \int 3365793 x d x + \int 531441 d x

\int 4096 x^{13} d x = \frac{2048 x ^{14}}{7}

\int 53248 x^{12} d x = 4096 x^{13}

\int 350208 x^{11} d x = 29184 x^{12}

\int 1509376 x^{10} d x = 137216 x^{11}

\int 4714240 x^{9} d x = 471424 x^{10}

\int 11204352 x^{8} d x = 1244928 x^{9}

\int 20782336 x^{7} d x = 2597792 x^{8}

\int 30405376 x^{6} d x = \frac{30405376 x ^{7}}{7}

\int 35070192 x^{5} d x = 5845032 x^{6}

\int 31512240 x^{4} d x = 6302448 x^{5}

\int 21479256 x^{3} d x = 5369814 x^{4}

\int 10550088 x^{2} d x = 3516696 x^{3}

\int 3365793 x d x = \frac{3365793 x ^{2}}{2}

\int 531441 d x = 531441 x

= \frac{2048 x ^{14}}{7} + 4096 x^{13} + 29184 x^{12} + 137216 x^{11} + 471424 x^{10} + 1244928 x^{9} + 2597792 x^{8} + \frac{30405376 x ^{7}}{7} + 5845032 x^{6} + 6302448 x^{5} + 5369814 x^{4} + 3516696 x^{3} + \frac{3365793 x ^{2}}{2} + 531441 x

解答补常数

= \frac{2048 x ^{14}}{7} + 4096 x^{13} + 29184 x^{12} + 137216 x^{11} + 471424 x^{10} + 1244928 x^{9} + 2597792 x^{8} + \frac{30405376 x ^{7}}{7} + 5845032 x^{6} + 6302448 x^{5} + 5369814 x^{4} + 3516696 x^{3} + \frac{3365793 x ^{2}}{2} + 531441 x + C

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