integral of ((sqrt(x)+2)^7)/(2sqrt(x))

解答

\int \frac{( x + 2 ) ^{7}}{2 x} d x

\frac{1}{2} (\frac{x ^{4}}{4} + 4 x^{\frac{7}{2}} + 28 x^{3} + 112 x^{\frac{5}{2}} + 280 x^{2} + 448 x^{\frac{3}{2}} + 448 x + 256 x) + C

求解步骤

\int \frac{( x + 2 ) ^{7}}{2 x} d x

提出常数:

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

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

长除

\frac{( x + 2 ) ^{7}}{x} : x^{3} + 14 (x)^{5} + 84 x^{2} + 280 (x)^{3} + 560 x + 672 x + 448 + \frac{128}{x}

= \frac{1}{2} \cdot \int x^{3} + 14 (x)^{5} + 84 x^{2} + 280 (x)^{3} + 560 x + 672 x + 448 + \frac{128}{x} 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}{2} (\int x^{3} d x + \int 14 (x)^{5} d x + \int 84 x^{2} d x + \int 280 (x)^{3} d x + \int 560 x d x + \int 672 x d x + \int 448 d x + \int \frac{128}{x} d x)

\int x^{3} d x = \frac{x ^{4}}{4}

\int 14 (x)^{5} d x = 4 x^{\frac{7}{2}}

\int 84 x^{2} d x = 28 x^{3}

\int 280 (x)^{3} d x = 112 x^{\frac{5}{2}}

\int 560 x d x = 280 x^{2}

\int 672 x d x = 448 x^{\frac{3}{2}}

\int 448 d x = 448 x

\int \frac{128}{x} d x = 256 x

= \frac{1}{2} (\frac{x ^{4}}{4} + 4 x^{\frac{7}{2}} + 28 x^{3} + 112 x^{\frac{5}{2}} + 280 x^{2} + 448 x^{\frac{3}{2}} + 448 x + 256 x)

解答补常数

= \frac{1}{2} (\frac{x ^{4}}{4} + 4 x^{\frac{7}{2}} + 28 x^{3} + 112 x^{\frac{5}{2}} + 280 x^{2} + 448 x^{\frac{3}{2}} + 448 x + 256 x) + C

\int 10 e^{x + 3} d x

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