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受欢迎的微积分 >

integral of sqrt(x)sqrt(0.04x^{3/4)+4}

解答

\int x 0.04 x^{\frac{3}{4}} + 4 d x

解答

833.33333 \dots (0.4 (0.04 x^{\frac{3}{4}} + 4)^{\frac{5}{2}} - 2.66666 \dots (0.04 x^{\frac{3}{4}} + 4)^{\frac{3}{2}}) + C

求解步骤

\int x 0.04 x^{\frac{3}{4}} + 4 d x

使用换元积分法

= \int \frac{5 \cdot 5 ^{\frac{1}{2}} u ( 3 25 ( u - 4 ) 3 u - 4 ) ^{\frac{3}{4}}}{0.03} d u

提出常数:

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

= \frac{5 \cdot 5 ^{\frac{1}{2}}}{0.03} \cdot \int u (325 (u - 4) 3 u - 4)^{\frac{3}{4}} d u

化简

= 372.67799 \dots \cdot \int u (325 (u - 4) 3 u - 4)^{\frac{3}{4}} d u

使用换元积分法

= 372.67799 \dots \cdot \int 3 \cdot 5^{\frac{1}{2}} v^{5} v^{3} + 4 d v

提出常数:

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

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \int v^{5} v^{3} + 4 d v

使用换元积分法

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \int \frac{( w - 4 ) w ^{\frac{1}{2}}}{3} d w

提出常数:

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

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \frac{1}{3} \cdot \int (w - 4) w^{\frac{1}{2}} d w

乘开

(w - 4) w^{\frac{1}{2}} : w^{\frac{3}{2}} - 4 w^{\frac{1}{2}}

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \frac{1}{3} \cdot \int w^{\frac{3}{2}} - 4 w^{\frac{1}{2}} d w

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

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \frac{1}{3} (\int w^{\frac{3}{2}} d w - \int 4 w^{\frac{1}{2}} d w)

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

\int 4 w^{\frac{1}{2}} d w = \frac{8}{3} w^{\frac{3}{2}}

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \frac{1}{3} (\frac{2}{5} w^{\frac{5}{2}} - \frac{8}{3} w^{\frac{3}{2}})

代回

= 372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \frac{1}{3} \frac{2}{5} ((3 0.04 x^{\frac{3}{4}} + 4 - 4)^{3} + 4)^{\frac{5}{2}} - \frac{8}{3} ((3 0.04 x^{\frac{3}{4}} + 4 - 4)^{3} + 4)^{\frac{3}{2}}

化简

372.67799 \dots \cdot 3 \cdot 5^{\frac{1}{2}} \cdot \frac{1}{3} \frac{2}{5} ((3 0.04 x^{\frac{3}{4}} + 4 - 4)^{3} + 4)^{\frac{5}{2}} - \frac{8}{3} ((3 0.04 x^{\frac{3}{4}} + 4 - 4)^{3} + 4)^{\frac{3}{2}} : 833.33333 \dots (0.4 (0.04 x^{\frac{3}{4}} + 4)^{\frac{5}{2}} - 2.66666 \dots (0.04 x^{\frac{3}{4}} + 4)^{\frac{3}{2}})

= 833.33333 \dots (0.4 (0.04 x^{\frac{3}{4}} + 4)^{\frac{5}{2}} - 2.66666 \dots (0.04 x^{\frac{3}{4}} + 4)^{\frac{3}{2}})

解答补常数

= 833.33333 \dots (0.4 (0.04 x^{\frac{3}{4}} + 4)^{\frac{5}{2}} - 2.66666 \dots (0.04 x^{\frac{3}{4}} + 4)^{\frac{3}{2}}) + C

作图

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