integral from 0 to 2 of (ln(1+x))/(1+x)

解答

\int_{0}^{2} \frac{ln ( 1 + x )}{1 + x} d x

\frac{ln ^{2} ( 3 )}{2}

十进制

0.60347 \dots

求解步骤

\int_{0}^{2} \frac{ln ( 1 + x )}{1 + x} d x

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

= \int_{0}^{2} ln (1 + x) \frac{1}{1 + x} d x

使用换元积分法

= \int_{0}^{l n (3)} u d u

使用幂法则

= [\frac{u ^{2}}{2}]_{0}^{l n (3)}

带入上下限计算后得:

\frac{ln ^{2} ( 3 )}{2}

= \frac{ln ^{2} ( 3 )}{2}

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