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受欢迎的微积分 >

integral of (x^2+3x-1)^4(2x+3)

解答

\int (x^{2} + 3 x - 1)^{4} (2 x + 3) d x

解答

\frac{1}{5120} (4 x^{2} + 12 x - 4)^{5} + C

求解步骤

\int (x^{2} + 3 x - 1)^{4} (2 x + 3) d x

使用换元积分法

= \int \frac{u ( u ^{2} - 13 ) ^{4}}{512} d u

提出常数:

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

= \frac{1}{512} \cdot \int u (u^{2} - 13)^{4} d u

使用换元积分法

= \frac{1}{512} \cdot \int \frac{v ^{4}}{2} d v

提出常数:

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

= \frac{1}{512} \cdot \frac{1}{2} \cdot \int v^{4} d v

使用幂法则

= \frac{1}{512} \cdot \frac{1}{2} \cdot \frac{v ^{5}}{5}

代回

= \frac{1}{512} \cdot \frac{1}{2} \cdot \frac{( ( 2 x + 3 ) ^{2} - 13 ) ^{5}}{5}

化简

\frac{1}{512} \cdot \frac{1}{2} \cdot \frac{( ( 2 x + 3 ) ^{2} - 13 ) ^{5}}{5} : \frac{1}{5120} (4 x^{2} + 12 x - 4)^{5}

= \frac{1}{5120} (4 x^{2} + 12 x - 4)^{5}

解答补常数

= \frac{1}{5120} (4 x^{2} + 12 x - 4)^{5} + C

作图

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