integral of (3x+1)^{11}

解答

\int (3 x + 1)^{11} d x

\frac{1}{36} (3 x + 1)^{12} + C

求解步骤

\int (3 x + 1)^{11} d x

使用换元积分法

= \int \frac{u ^{11}}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int u^{11} d u

使用幂法则

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

u = 3 x + 1

代回

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

化简

\frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{12}}{12} : \frac{1}{36} (3 x + 1)^{12}

= \frac{1}{36} (3 x + 1)^{12}

解答补常数

= \frac{1}{36} (3 x + 1)^{12} + C

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