integral of x(x^2-sqrt(3))^5

解答

\int x (x^{2} - 3)^{5} d x

\frac{x ^{12}}{12} - \frac{3 x ^{10}}{2} + \frac{15 x ^{8}}{4} - 53 x^{6} + \frac{45 x ^{4}}{4} - \frac{9 3 x ^{2}}{2} + C

求解步骤

\int x (x^{2} - 3)^{5} d x

乘开

x (x^{2} - 3)^{5} : x^{11} - 53 x^{9} + 30 x^{7} - 10 \cdot 3^{\frac{3}{2}} x^{5} + 45 x^{3} - 3^{\frac{5}{2}} x

= \int x^{11} - 53 x^{9} + 30 x^{7} - 10 \cdot 3^{\frac{3}{2}} x^{5} + 45 x^{3} - 3^{\frac{5}{2}} x 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 x^{11} d x - \int 53 x^{9} d x + \int 30 x^{7} d x - \int 10 \cdot 3^{\frac{3}{2}} x^{5} d x + \int 45 x^{3} d x - \int 3^{\frac{5}{2}} x d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 53 x^{9} d x = \frac{3 x ^{10}}{2}

\int 30 x^{7} d x = \frac{15 x ^{8}}{4}

\int 10 \cdot 3^{\frac{3}{2}} x^{5} d x = 53 x^{6}

\int 45 x^{3} d x = \frac{45 x ^{4}}{4}

\int 3^{\frac{5}{2}} x d x = \frac{9 3 x ^{2}}{2}

= \frac{x ^{12}}{12} - \frac{3 x ^{10}}{2} + \frac{15 x ^{8}}{4} - 53 x^{6} + \frac{45 x ^{4}}{4} - \frac{9 3 x ^{2}}{2}

解答补常数

= \frac{x ^{12}}{12} - \frac{3 x ^{10}}{2} + \frac{15 x ^{8}}{4} - 53 x^{6} + \frac{45 x ^{4}}{4} - \frac{9 3 x ^{2}}{2} + C

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