题目
int dfrac ({x)^3}({(1+{x)^2)}^2}dx=A.int dfrac ({x)^3}({(1+{x)^2)}^2}dx=B.int dfrac ({x)^3}({(1+{x)^2)}^2}dx=C.int dfrac ({x)^3}({(1+{x)^2)}^2}dx=D.int dfrac ({x)^3}({(1+{x)^2)}^2}dx=
A.
B.
C.
D.
题目解答
答案
选D
解析
步骤 1：将积分式子进行变形
$\int \dfrac {{x}^{3}}{{(1+{x}^{2})}^{2}}dx$
$=\dfrac {1}{2}\int \dfrac {{x}^{2}}{{(1+{x}^{2})}^{2}}d(1+{x}^{2})$
步骤 2：将分子进行拆分
$=\dfrac {1}{2}\int \dfrac {1+{x}^{2}-1}{{(1+{x}^{2})}^{2}}d(1+{x}^{2})$
步骤 3：将积分式子拆分为两个积分
$=\dfrac {1}{2}\int \dfrac {1}{1+{x}^{2}}d(1+{x}^{2})-\dfrac {1}{2}\int \dfrac {1}{{(1+{x}^{2})}^{2}}d(1+{x}^{2})$
步骤 4：分别计算两个积分
$=\dfrac {1}{2}[ \ln (1+{x}^{2})+\dfrac {1}{1+{x}^{2}}] +C$
步骤 5：整理得到最终结果
$=\dfrac {1}{2(1+{x}^{2})}+\dfrac {1}{2}\ln (1+{x}^{2})+C$
$\int \dfrac {{x}^{3}}{{(1+{x}^{2})}^{2}}dx$
$=\dfrac {1}{2}\int \dfrac {{x}^{2}}{{(1+{x}^{2})}^{2}}d(1+{x}^{2})$
步骤 2：将分子进行拆分
$=\dfrac {1}{2}\int \dfrac {1+{x}^{2}-1}{{(1+{x}^{2})}^{2}}d(1+{x}^{2})$
步骤 3：将积分式子拆分为两个积分
$=\dfrac {1}{2}\int \dfrac {1}{1+{x}^{2}}d(1+{x}^{2})-\dfrac {1}{2}\int \dfrac {1}{{(1+{x}^{2})}^{2}}d(1+{x}^{2})$
步骤 4：分别计算两个积分
$=\dfrac {1}{2}[ \ln (1+{x}^{2})+\dfrac {1}{1+{x}^{2}}] +C$
步骤 5：整理得到最终结果
$=\dfrac {1}{2(1+{x}^{2})}+\dfrac {1}{2}\ln (1+{x}^{2})+C$