integral of x^3(x^4-5)^4

解答

\int x^{3} (x^{4} - 5)^{4} d x

\frac{1}{20} (x^{4} - 5)^{5} + C

求解步骤

\int x^{3} (x^{4} - 5)^{4} d x

使用换元积分法

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

使用幂法则

= \frac{1}{4} \cdot \frac{u ^{5}}{5}

u = x^{4} - 5

代回

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

化简

\frac{1}{4} \cdot \frac{( x ^{4} - 5 ) ^{5}}{5} : \frac{1}{20} (x^{4} - 5)^{5}

= \frac{1}{20} (x^{4} - 5)^{5}

解答补常数

= \frac{1}{20} (x^{4} - 5)^{5} + C

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