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受欢迎的微积分 >

integral of x^{11}sqrt(x^6+6)

解答

\int x^{11} x^{6} + 6 d x

解答

\frac{1}{3} (\frac{1}{5} (x^{6} + 6)^{\frac{5}{2}} - 2 (x^{6} + 6)^{\frac{3}{2}}) + C

求解步骤

\int x^{11} x^{6} + 6 d x

使用换元积分法

= \int \frac{( u ^{2} - 6 ) u ^{2}}{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^{2} - 6) u^{2} d u

乘开

(u^{2} - 6) u^{2} : u^{4} - 6 u^{2}

= \frac{1}{3} \cdot \int u^{4} - 6 u^{2} 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^{4} d u - \int 6 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 6 u^{2} d u = 2 u^{3}

= \frac{1}{3} (\frac{u ^{5}}{5} - 2 u^{3})

u = x^{6} + 6

代回

= \frac{1}{3} (\frac{( x ^{6} + 6 ) ^{5}}{5} - 2 (x^{6} + 6)^{3})

化简

\frac{1}{3} (\frac{( x ^{6} + 6 ) ^{5}}{5} - 2 (x^{6} + 6)^{3}) : \frac{1}{3} (\frac{1}{5} (x^{6} + 6)^{\frac{5}{2}} - 2 (x^{6} + 6)^{\frac{3}{2}})

= \frac{1}{3} (\frac{1}{5} (x^{6} + 6)^{\frac{5}{2}} - 2 (x^{6} + 6)^{\frac{3}{2}})

解答补常数

= \frac{1}{3} (\frac{1}{5} (x^{6} + 6)^{\frac{5}{2}} - 2 (x^{6} + 6)^{\frac{3}{2}}) + C

作图

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