integral from 0 to 15 of x/((1+x)^{3/4)}

解答

\int_{0}^{15} \frac{x}{( 1 + x ) ^{\frac{3}{4}}} d x

\frac{104}{5}

十进制

20.8

求解步骤

\int_{0}^{15} \frac{x}{( 1 + x ) ^{\frac{3}{4}}} d x

使用换元积分法

= \int_{1}^{16} \frac{u - 1}{u ^{\frac{3}{4}}} d u

乘开

\frac{u - 1}{u ^{\frac{3}{4}}} : u^{\frac{1}{4}} - \frac{1}{u ^{\frac{3}{4}}}

= \int_{1}^{16} u^{\frac{1}{4}} - \frac{1}{u ^{\frac{3}{4}}} 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_{1}^{16} u^{\frac{1}{4}} d u - \int_{1}^{16} \frac{1}{u ^{\frac{3}{4}}} d u

\int_{1}^{16} u^{\frac{1}{4}} d u = \frac{124}{5}

\int_{1}^{16} \frac{1}{u ^{\frac{3}{4}}} d u = 4

= \frac{124}{5} - 4

化简

\frac{124}{5} - 4 : \frac{104}{5}

= \frac{104}{5}

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