integral of x^3sqrt(x^2-25)

解答

\int x^{3} x^{2} - 25 d x

\frac{1}{5} (x^{2} - 25)^{\frac{5}{2}} + \frac{25}{3} (x^{2} - 25)^{\frac{3}{2}} + C

求解步骤

\int x^{3} x^{2} - 25 d x

使用换元积分法

= \int (u^{2} + 25) u^{2} d u

乘开

(u^{2} + 25) u^{2} : u^{4} + 25 u^{2}

= \int u^{4} + 25 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

= \int u^{4} d u + \int 25 u^{2} d u

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

\int 25 u^{2} d u = \frac{25 u ^{3}}{3}

= \frac{u ^{5}}{5} + \frac{25 u ^{3}}{3}

u = x^{2} - 25

代回

= \frac{( x ^{2} - 25 ) ^{5}}{5} + \frac{25 ( x ^{2} - 25 ) ^{3}}{3}

化简

\frac{( x ^{2} - 25 ) ^{5}}{5} + \frac{25 ( x ^{2} - 25 ) ^{3}}{3} : \frac{1}{5} (x^{2} - 25)^{\frac{5}{2}} + \frac{25}{3} (x^{2} - 25)^{\frac{3}{2}}

= \frac{1}{5} (x^{2} - 25)^{\frac{5}{2}} + \frac{25}{3} (x^{2} - 25)^{\frac{3}{2}}

解答补常数

= \frac{1}{5} (x^{2} - 25)^{\frac{5}{2}} + \frac{25}{3} (x^{2} - 25)^{\frac{3}{2}} + C

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