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受欢迎的微积分 >

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

解答

\int \frac{3 x}{1 + x ^{3}} d x

解答

2 arctan (x^{\frac{3}{2}}) + C

求解步骤

\int \frac{3 x}{1 + x ^{3}} d x

提出常数:

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

= 3 \cdot \int \frac{x}{1 + x ^{3}} d x

使用换元积分法

= 3 \cdot \int \frac{2 u ^{2}}{1 + u ^{6}} d u

提出常数:

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

= 3 \cdot 2 \cdot \int \frac{u ^{2}}{1 + u ^{6}} d u

使用换元积分法

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

提出常数:

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

= 3 \cdot 2 \cdot \frac{1}{3} \cdot \int \frac{1}{1 + v ^{2}} d v

使用积分常见定则:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= 3 \cdot 2 \cdot \frac{1}{3} arctan (v)

代回

= 3 \cdot 2 \cdot \frac{1}{3} arctan ((x)^{3})

化简

3 \cdot 2 \cdot \frac{1}{3} arctan ((x)^{3}) : 2 arctan (x^{\frac{3}{2}})

= 2 arctan (x^{\frac{3}{2}})

解答补常数

= 2 arctan (x^{\frac{3}{2}}) + C

作图

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derivative of 1/(1+e^x)

\frac{d}{d x} (\frac{1}{1 + e ^{x}})

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\frac{d}{d x} (\frac{sin ( x + 1 )}{e ^{x}})

inverselaplace (e^{-4x})/((x-2)^2)

in v erse l a pl a ce \frac{e ^{- 4 x}}{( x - 2 ) ^{2}}

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x \to \infty lim ((1 + \frac{c}{x})^{2 x})