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受欢迎的微积分 >

integral of (3x^3)/(4+x^8)

解答

\int \frac{3 x ^{3}}{4 + x ^{8}} d x

解答

\frac{3}{8} arctan (\frac{x ^{4}}{2}) + C

求解步骤

\int \frac{3 x ^{3}}{4 + x ^{8}} 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 ^{3}}{4 + x ^{8}} d x

使用换元积分法

= 3 \cdot \int \frac{1}{4 ( 4 + u ^{2} )} d u

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{4 + u ^{2}} d u

使用换元积分法

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{2 ( v ^{2} + 1 )} d v

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot \frac{1}{2} \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)

= 3 \cdot \frac{1}{4} \cdot \frac{1}{2} arctan (v)

代回

= 3 \cdot \frac{1}{4} \cdot \frac{1}{2} arctan (\frac{x ^{4}}{2})

化简

3 \cdot \frac{1}{4} \cdot \frac{1}{2} arctan (\frac{x ^{4}}{2}) : \frac{3}{8} arctan (\frac{x ^{4}}{2})

= \frac{3}{8} arctan (\frac{x ^{4}}{2})

解答补常数

= \frac{3}{8} arctan (\frac{x ^{4}}{2}) + C

作图

流行的例子

y^{''}-3y^'+2y=e^{3x}(x+1)

y^{^{''}} - 3 y^{^{'}} + 2 y = e^{3 x} (x + 1)

tangent f(x)=9x-8sqrt(x),(1,1)

t an g e n t f (x) = 9 x - 8 x, (1, 1)

integral of (e^{3x})/(e^{3x)-e^{-3x}}

\int \frac{e ^{3 x}}{e ^{3 x} - e ^{- 3 x}} d x

limit as x approaches-6 of (-9x)/(x+6)

x \to - 6 lim (\frac{- 9 x}{x + 6})

integral of 9sec^3(x)

\int 9 sec^{3} (x) d x