integral of (4x^3-2)^2*x^2

解答

\int (4 x^{3} - 2)^{2} \cdot x^{2} d x

\frac{1}{36} (4 x^{3} - 2)^{3} + C

求解步骤

\int (4 x^{3} - 2)^{2} x^{2} d x

使用换元积分法

= \int \frac{u ^{2}}{12} d u

提出常数:

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

= \frac{1}{12} \cdot \int u^{2} d u

使用幂法则

= \frac{1}{12} \cdot \frac{u ^{3}}{3}

u = 4 x^{3} - 2

代回

= \frac{1}{12} \cdot \frac{( 4 x ^{3} - 2 ) ^{3}}{3}

化简

\frac{1}{12} \cdot \frac{( 4 x ^{3} - 2 ) ^{3}}{3} : \frac{1}{36} (4 x^{3} - 2)^{3}

= \frac{1}{36} (4 x^{3} - 2)^{3}

解答补常数

= \frac{1}{36} (4 x^{3} - 2)^{3} + C

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