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受欢迎的微积分 >

integral of 1/((2x+2)(ln(x+1))^2)

解答

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

解答

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

求解步骤

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

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

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

使用换元积分法

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

使用换元积分法

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

使用幂法则

= \frac{1}{2} (- \frac{1}{v})

代回

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

化简

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

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

解答补常数

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

作图

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