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受欢迎的微积分 >

integral of x/(1+4x^4)

解答

\int \frac{x}{1 + 4 x ^{4}} d x

解答

\frac{1}{4} arctan (2 x^{2}) + C

求解步骤

\int \frac{x}{1 + 4 x ^{4}} d x

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

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

代回

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

化简

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

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

解答补常数

= \frac{1}{4} arctan (2 x^{2}) + C

作图

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