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受欢迎的微积分 >

integral of (sqrt(2x)+1)^3sqrt(2)

解答

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

解答

\frac{2}{5} (42 x^{\frac{5}{2}} + 15 x^{2} + 102 x^{\frac{3}{2}} + 5 x) + C

求解步骤

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

提出常数:

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

= 2 \cdot \int (2 x + 1)^{3} d x

使用换元积分法

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

提出常数:

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

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

乘开

(u + 1)^{3} : u^{\frac{3}{2}} + 3 u + 3 u + 1

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

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

\int 1 d u = u

= 2 \frac{1}{2} (\frac{2}{5} u^{\frac{5}{2}} + \frac{3 u ^{2}}{2} + 2 u^{\frac{3}{2}} + u)

u = 2 x

代回

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

化简

2 \frac{1}{2} (\frac{2}{5} (2 x)^{\frac{5}{2}} + \frac{3 ( 2 x ) ^{2}}{2} + 2 (2 x)^{\frac{3}{2}} + 2 x) : \frac{2}{5} (42 x^{\frac{5}{2}} + 15 x^{2} + 102 x^{\frac{3}{2}} + 5 x)

= \frac{2}{5} (42 x^{\frac{5}{2}} + 15 x^{2} + 102 x^{\frac{3}{2}} + 5 x)

解答补常数

= \frac{2}{5} (42 x^{\frac{5}{2}} + 15 x^{2} + 102 x^{\frac{3}{2}} + 5 x) + C

作图

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