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受欢迎的微积分 >

integral of (sqrt(x)-1)^3

解答

\int (x - 1)^{3} d x

解答

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

求解步骤

\int (x - 1)^{3} d x

使用换元积分法

= \int (u - 1)^{3} \cdot 2 u d u

提出常数:

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

= 2 \cdot \int (u - 1)^{3} u d u

乘开

(u - 1)^{3} u : u^{4} - 3 u^{3} + 3 u^{2} - u

= 2 \cdot \int u^{4} - 3 u^{3} + 3 u^{2} - u 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

= 2 (\int u^{4} d u - \int 3 u^{3} d u + \int 3 u^{2} d u - \int u d u)

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

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

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

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

= 2 (\frac{u ^{5}}{5} - \frac{3 u ^{4}}{4} + u^{3} - \frac{u ^{2}}{2})

u = x

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of (sin(1/(x^8)))/(x^9)

\int \frac{sin ( \frac{1}{x ^{8}} )}{x ^{9}} d x

integral of-5x^3e^{x^4}

\int - 5 x^{3} e^{x^{4}} d x

integral of (1+e^x)/(1+e^{2x)}

\int \frac{1 + e ^{x}}{1 + e ^{2 x}} d x

integral of x/(sqrt(5x^2-2x+1))

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

integral of 1/(v^3)

\int \frac{1}{v ^{3}} d v