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受欢迎的微积分 >

integral of (x^7)/(x^4+1)

解答

\int \frac{x ^{7}}{x ^{4} + 1} d x

解答

\frac{1}{4} (x^{4} + 1 - ln x^{4} + 1) + C

求解步骤

\int \frac{x ^{7}}{x ^{4} + 1} d x

使用换元积分法

= \int \frac{u ^{\frac{1}{7}}}{7 ( u ^{\frac{4}{7}} + 1 )} d u

提出常数:

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

= \frac{1}{7} \cdot \int \frac{u ^{\frac{1}{7}}}{u ^{\frac{4}{7}} + 1} d u

使用换元积分法

= \frac{1}{7} \cdot \int \frac{7 ( v - 1 )}{4 v} d v

提出常数:

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

= \frac{1}{7} \cdot \frac{7}{4} \cdot \int \frac{v - 1}{v} d v

乘开

\frac{v - 1}{v} : 1 - \frac{1}{v}

= \frac{1}{7} \cdot \frac{7}{4} \cdot \int 1 - \frac{1}{v} d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{7} \cdot \frac{7}{4} (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= \frac{1}{7} \cdot \frac{7}{4} (v - ln ∣ v ∣)

代回

= \frac{1}{7} \cdot \frac{7}{4} ((x^{7})^{\frac{4}{7}} + 1 - ln (x^{7})^{\frac{4}{7}} + 1)

化简

\frac{1}{7} \cdot \frac{7}{4} ((x^{7})^{\frac{4}{7}} + 1 - ln (x^{7})^{\frac{4}{7}} + 1) : \frac{1}{4} (x^{4} + 1 - ln x^{4} + 1)

= \frac{1}{4} (x^{4} + 1 - ln x^{4} + 1)

解答补常数

= \frac{1}{4} (x^{4} + 1 - ln x^{4} + 1) + C

作图

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