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受欢迎的微积分 >

integral of-4ln(2x+1)

解答

\int - 4 ln (2 x + 1) d x

解答

- 2 (2 x ln (2 x + 1) + ln (2 x + 1) - 2 x - 1) + C

求解步骤

\int - 4 ln (2 x + 1) d x

提出常数:

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

= - 4 \cdot \int ln (2 x + 1) d x

使用换元积分法

= - 4 \cdot \int ln (u) \frac{1}{2} d u

提出常数:

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

= - 4 \cdot \frac{1}{2} \cdot \int ln (u) d u

使用分布积分法

= - 4 \cdot \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= - 4 \cdot \frac{1}{2} (u ln (u) - u)

u = 2 x + 1

代回

= - 4 \cdot \frac{1}{2} ((2 x + 1) ln (2 x + 1) - (2 x + 1))

化简

- 4 \cdot \frac{1}{2} ((2 x + 1) ln (2 x + 1) - (2 x + 1)) : - 2 (2 x ln (2 x + 1) + ln (2 x + 1) - 2 x - 1)

= - 2 (2 x ln (2 x + 1) + ln (2 x + 1) - 2 x - 1)

解答补常数

= - 2 (2 x ln (2 x + 1) + ln (2 x + 1) - 2 x - 1) + C

作图

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