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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

= \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

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

使用换元积分法

= 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

= 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)

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

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

(d^2y)/(dx^2)+2(dy)/(dx)-y=20xe^x

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} - y = 20 x e^{x}

\frac{d y}{d x} + y = x

normal f(x)=2x^2-5,(2,-2)

n or ma l f (x) = 2 x^{2} - 5, (2, - 2)

derivative of ln(x+ln(sin(y)))

\frac{d}{d x} (ln (x) + ln (sin (y)))

derivative of-7x

\frac{d}{d x} (- 7 x)