integral of ((4x^m-3x^n)^3)/(sqrt(x))

解答

\int \frac{( 4 x ^{m} - 3 x ^{n} ) ^{3}}{x} d x

\frac{128 x ^{\frac{6 m + 1}{2}}}{6 m + 1} - \frac{288 x ^{\frac{4 m + 2 n + 1}{2}}}{4 m + 2 n + 1} + \frac{216 x ^{\frac{2 m + 4 n + 1}{2}}}{2 m + 4 n + 1} - \frac{54 x ^{\frac{6 n + 1}{2}}}{6 n + 1} + C

求解步骤

\int \frac{( 4 x ^{m} - 3 x ^{n} ) ^{3}}{x} d x

乘开

\frac{( 4 x ^{m} - 3 x ^{n} ) ^{3}}{x} : 64 x^{\frac{6 m - 1}{2}} - 144 x^{\frac{4 m + 2 n - 1}{2}} + 108 x^{\frac{2 m + 4 n - 1}{2}} - 27 x^{\frac{6 n - 1}{2}}

= \int 64 x^{\frac{6 m - 1}{2}} - 144 x^{\frac{4 m + 2 n - 1}{2}} + 108 x^{\frac{2 m + 4 n - 1}{2}} - 27 x^{\frac{6 n - 1}{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

= \int 64 x^{\frac{6 m - 1}{2}} d x - \int 144 x^{\frac{4 m + 2 n - 1}{2}} d x + \int 108 x^{\frac{2 m + 4 n - 1}{2}} d x - \int 27 x^{\frac{6 n - 1}{2}} d x

\int 64 x^{\frac{6 m - 1}{2}} d x = \frac{128 x ^{\frac{6 m + 1}{2}}}{6 m + 1}

\int 144 x^{\frac{4 m + 2 n - 1}{2}} d x = \frac{288 x ^{\frac{4 m + 2 n + 1}{2}}}{4 m + 2 n + 1}

\int 108 x^{\frac{2 m + 4 n - 1}{2}} d x = \frac{216 x ^{\frac{2 m + 4 n + 1}{2}}}{2 m + 4 n + 1}

\int 27 x^{\frac{6 n - 1}{2}} d x = \frac{54 x ^{\frac{6 n + 1}{2}}}{6 n + 1}

= \frac{128 x ^{\frac{6 m + 1}{2}}}{6 m + 1} - \frac{288 x ^{\frac{4 m + 2 n + 1}{2}}}{4 m + 2 n + 1} + \frac{216 x ^{\frac{2 m + 4 n + 1}{2}}}{2 m + 4 n + 1} - \frac{54 x ^{\frac{6 n + 1}{2}}}{6 n + 1}

解答补常数

= \frac{128 x ^{\frac{6 m + 1}{2}}}{6 m + 1} - \frac{288 x ^{\frac{4 m + 2 n + 1}{2}}}{4 m + 2 n + 1} + \frac{216 x ^{\frac{2 m + 4 n + 1}{2}}}{2 m + 4 n + 1} - \frac{54 x ^{\frac{6 n + 1}{2}}}{6 n + 1} + C

x \to \infty lim ((- 1)^{- x})

\int (x^{4} - x^{2} + \frac{1}{4 x ^{3}} - \frac{1}{x ^{2}}) d x

in v erse l a pl a ce \frac{3}{( s ^{2} + 4 )}

d er i v a t i v e 27 n + 3 q^{2} n

d er i v a t i v e y = \frac{4 x - 1}{2 x + 5}