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受欢迎的微积分 >

integral of (7sqrt(x))/(x^3+1)

解答

\int \frac{7 x}{x ^{3} + 1} d x

解答

\frac{14}{3} arctan (x^{\frac{3}{2}}) + C

求解步骤

\int \frac{7 x}{x ^{3} + 1} d x

提出常数:

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

= 7 \cdot \int \frac{x}{x ^{3} + 1} d x

使用换元积分法

= 7 \cdot \int \frac{2 u ^{2}}{u ^{6} + 1} d u

提出常数:

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

= 7 \cdot 2 \cdot \int \frac{u ^{2}}{u ^{6} + 1} d u

使用换元积分法

= 7 \cdot 2 \cdot \int \frac{1}{3 ( v ^{2} + 1 )} d v

提出常数:

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

= 7 \cdot 2 \cdot \frac{1}{3} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

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

= 7 \cdot 2 \cdot \frac{1}{3} arctan (v)

代回

= 7 \cdot 2 \cdot \frac{1}{3} arctan ((x)^{3})

化简

7 \cdot 2 \cdot \frac{1}{3} arctan ((x)^{3}) : \frac{14}{3} arctan (x^{\frac{3}{2}})

= \frac{14}{3} arctan (x^{\frac{3}{2}})

解答补常数

= \frac{14}{3} arctan (x^{\frac{3}{2}}) + C

作图

流行的例子

derivative y=(x+7)(7sqrt(x)+2)

d er i v a t i v e y = (x + 7) (7 x + 2)

derivative of x^3(8-3x)

\frac{d}{d x} (x^{3} (8 - 3 x))

derivative of (1+6x^2(x-x^2))

\frac{d}{d x} ((1 + 6 x^{2}) (x - x^{2}))

inverselaplace (s+1)e^{-spi}

in v erse l a pl a ce (s + 1) e^{- s π}

(dy)/(dx)=(x^2)

\frac{d y}{d x} = (x^{2})