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受欢迎的微积分 >

integral of (x^2-2x+6)(2x-2)

解答

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

解答

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

求解步骤

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

(x^{2} - 2 x + 6) (2 x - 2) = 2 (x^{2} - 2 x + 6) (x - 1)

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

提出常数:

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

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

使用换元积分法

= 2 \cdot \int \frac{u}{2} d u

提出常数:

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

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

使用幂法则

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{2}}{2}

u = x^{2} - 2 x + 6

代回

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

化简

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 2 x + 6 ) ^{2}}{2} : \frac{1}{2} (x^{2} - 2 x + 6)^{2}

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

解答补常数

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

作图

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