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受欢迎的微积分 >

integral of x(3x^2-1)^3

解答

\int x (3 x^{2} - 1)^{3} d x

解答

\frac{1}{24} (3 x^{2} - 1)^{4} + C

求解步骤

\int x (3 x^{2} - 1)^{3} d x

使用换元积分法

= \int \frac{u ^{3}}{6} d u

提出常数:

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

= \frac{1}{6} \cdot \int u^{3} d u

使用幂法则

= \frac{1}{6} \cdot \frac{u ^{4}}{4}

u = 3 x^{2} - 1

代回

= \frac{1}{6} \cdot \frac{( 3 x ^{2} - 1 ) ^{4}}{4}

化简

\frac{1}{6} \cdot \frac{( 3 x ^{2} - 1 ) ^{4}}{4} : \frac{1}{24} (3 x^{2} - 1)^{4}

= \frac{1}{24} (3 x^{2} - 1)^{4}

解答补常数

= \frac{1}{24} (3 x^{2} - 1)^{4} + C

作图

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\int \frac{12 x ^{2} + 16 x + 3}{4 x ^{3} + 8 x ^{2} + 3 x} d x

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积分 (x^2+3)(x-1)

in t e g r a l (x^{2} + 3) (x - 1)