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受欢迎的微积分 >

integral of 3x(6x^2+1)^3

解答

\int 3 x (6 x^{2} + 1)^{3} d x

解答

\frac{1}{16} (6 x^{2} + 1)^{4} + C

求解步骤

\int 3 x (6 x^{2} + 1)^{3} d x

提出常数:

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

= 3 \cdot \int x (6 x^{2} + 1)^{3} d x

使用换元积分法

= 3 \cdot \int \frac{u ^{3}}{12} d u

提出常数:

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

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

使用幂法则

= 3 \cdot \frac{1}{12} \cdot \frac{u ^{4}}{4}

u = 6 x^{2} + 1

代回

= 3 \cdot \frac{1}{12} \cdot \frac{( 6 x ^{2} + 1 ) ^{4}}{4}

化简

3 \cdot \frac{1}{12} \cdot \frac{( 6 x ^{2} + 1 ) ^{4}}{4} : \frac{1}{16} (6 x^{2} + 1)^{4}

= \frac{1}{16} (6 x^{2} + 1)^{4}

解答补常数

= \frac{1}{16} (6 x^{2} + 1)^{4} + C

作图

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