integral of (sqrt(x)(x+1))/(4x^4)

解答

\int \frac{x ( x + 1 )}{4 x ^{4}} d x

- \frac{5 x + 3}{30 x ^{\frac{5}{2}}} + C

求解步骤

\int \frac{x ( x + 1 )}{4 x ^{4}} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{x ( x + 1 )}{x ^{4}} d x

乘开

\frac{x ( x + 1 )}{x ^{4}} : \frac{1}{x ^{\frac{5}{2}}} + \frac{1}{x ^{\frac{7}{2}}}

= \frac{1}{4} \cdot \int \frac{1}{x ^{\frac{5}{2}}} + \frac{1}{x ^{\frac{7}{2}}} 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

= \frac{1}{4} (\int \frac{1}{x ^{\frac{5}{2}}} d x + \int \frac{1}{x ^{\frac{7}{2}}} d x)

\int \frac{1}{x ^{\frac{5}{2}}} d x = - \frac{2}{3 x ^{\frac{3}{2}}}

\int \frac{1}{x ^{\frac{7}{2}}} d x = - \frac{2}{5 x ^{\frac{5}{2}}}

= \frac{1}{4} (- \frac{2}{3 x ^{\frac{3}{2}}} - \frac{2}{5 x ^{\frac{5}{2}}})

化简

\frac{1}{4} (- \frac{2}{3 x ^{\frac{3}{2}}} - \frac{2}{5 x ^{\frac{5}{2}}}) : - \frac{5 x + 3}{30 x ^{\frac{5}{2}}}

= - \frac{5 x + 3}{30 x ^{\frac{5}{2}}}

解答补常数

= - \frac{5 x + 3}{30 x ^{\frac{5}{2}}} + C

\frac{\partial}{\partial x} (y ln (x + y))

\int 16 e^{- 16 x} d x

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t \to 0 lim (f (t))