1.求下列不定积分:-|||-(1) int dfrac ({x)^3}(x-1)dx;-|||-(2) int dfrac (x-2)({x)^2-7x+12}dx;-|||-(3) int dfrac (dx)(1+{x)^3};-|||-(4) int dfrac (dx)(1+{x)^4};-|||-(5) int dfrac (dx)((x-1){({x)^2+1)}^2} ;-|||-(6) int dfrac (x-2)({(2{x)^2+2x+1)}^2}dx

(1) $\int \dfrac {{x}^{3}}{x-1}dx$
步骤 1：将被积函数进行多项式除法
$\dfrac {{x}^{3}}{x-1} = {x}^{2} + x + 1 + \dfrac {1}{x-1}$
步骤 2：对每一项进行积分
$\int \dfrac {{x}^{3}}{x-1}dx = \int ({x}^{2} + x + 1 + \dfrac {1}{x-1})dx$
步骤 3：计算积分
$\int ({x}^{2} + x + 1 + \dfrac {1}{x-1})dx = \dfrac {1}{3}{x}^{3} + \dfrac {1}{2}{x}^{2} + x + \ln |x-1| + C$
【答案】
$\dfrac {1}{3}{x}^{3} + \dfrac {1}{2}{x}^{2} + x + \ln |x-1| + C$

(2) $\int \dfrac {x-2}{{x}^{2}-7x+12}dx$
步骤 1：将分母分解因式
${x}^{2}-7x+12 = (x-3)(x-4)$
步骤 2：将被积函数进行部分分式分解
$\dfrac {x-2}{(x-3)(x-4)} = \dfrac {2}{x-4} - \dfrac {1}{x-3}$
步骤 3：对每一项进行积分
$\int (\dfrac {2}{x-4} - \dfrac {1}{x-3})dx = 2\ln |x-4| - \ln |x-3| + C$
【答案】
$2\ln |x-4| - \ln |x-3| + C$

(3) $\int \dfrac {dx}{1+{x}^{3}}$
步骤 1：将分母分解因式
$1+{x}^{3} = (1+x)({x}^{2}-x+1)$
步骤 2：将被积函数进行部分分式分解
$\dfrac {1}{(1+x)({x}^{2}-x+1)} = \dfrac {1}{3}(\dfrac {1}{1+x} + \dfrac {2-x}{{x}^{2}-x+1})$
步骤 3：对每一项进行积分
$\int \dfrac {1}{3}(\dfrac {1}{1+x} + \dfrac {2-x}{{x}^{2}-x+1})dx = \dfrac {1}{3}\ln |x+1| - \dfrac {1}{6}\int \dfrac {2x-1}{{x}^{2}-x+1}dx + \dfrac {1}{2}\int \dfrac {dx}{{(x-\dfrac {1}{2})}^{2}+\dfrac {3}{4}}$
步骤 4：计算积分
$\dfrac {1}{3}\ln |x+1| - \dfrac {1}{6}\ln |{x}^{2}-x+1| + \dfrac {\sqrt {3}}{3}\arctan \dfrac {2x-1}{\sqrt {3}} + C$