integral of x^6(x^7-3)^{3/4}

解答

\int x^{6} (x^{7} - 3)^{\frac{3}{4}} d x

\frac{4}{49} (x^{7} - 3)^{\frac{7}{4}} + C

求解步骤

\int x^{6} (x^{7} - 3)^{\frac{3}{4}} d x

使用换元积分法

= \int \frac{u ^{\frac{3}{4}}}{7} 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 u^{\frac{3}{4}} d u

使用幂法则

= \frac{1}{7} \cdot \frac{4}{7} u^{\frac{7}{4}}

u = x^{7} - 3

代回

= \frac{1}{7} \cdot \frac{4}{7} (x^{7} - 3)^{\frac{7}{4}}

化简

\frac{1}{7} \cdot \frac{4}{7} (x^{7} - 3)^{\frac{7}{4}} : \frac{4}{49} (x^{7} - 3)^{\frac{7}{4}}

= \frac{4}{49} (x^{7} - 3)^{\frac{7}{4}}

解答补常数

= \frac{4}{49} (x^{7} - 3)^{\frac{7}{4}} + C

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