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受欢迎的微积分 >

integral of ln(sqrt(x+2))

解答

\int ln (x + 2) d x

解答

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

求解步骤

\int ln (x + 2) d x

化简

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

= \int \frac{1}{2} ln (x + 2) d x

提出常数:

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

= \frac{1}{2} \cdot \int ln (x + 2) d x

使用换元积分法

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

使用分布积分法

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

\int 1 d u = u

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

u = x + 2

代回

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

化简

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

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

解答补常数

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

作图

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