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受欢迎的微积分 >

integral of (x^3)/(x^8+5)

解答

\int \frac{x ^{3}}{x ^{8} + 5} d x

解答

\frac{1}{4 5} arctan (\frac{x ^{4}}{5}) + C

求解步骤

\int \frac{x ^{3}}{x ^{8} + 5} d x

使用换元积分法

= \int \frac{1}{4 ( u ^{2} + 5 )} d u

提出常数:

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

= \frac{1}{4} \cdot \int \frac{1}{u ^{2} + 5} d u

使用换元积分法

= \frac{1}{4} \cdot \int \frac{1}{5 ( v ^{2} + 1 )} d v

提出常数:

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

= \frac{1}{4} \cdot \frac{1}{5} \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)

= \frac{1}{4} \cdot \frac{1}{5} arctan (v)

代回

= \frac{1}{4} \cdot \frac{1}{5} arctan (\frac{x ^{4}}{5})

化简

\frac{1}{4} \cdot \frac{1}{5} arctan (\frac{x ^{4}}{5}) : \frac{1}{4 5} arctan (\frac{x ^{4}}{5})

= \frac{1}{4 5} arctan (\frac{x ^{4}}{5})

解答补常数

= \frac{1}{4 5} arctan (\frac{x ^{4}}{5}) + C

作图

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