$$$\left(x^{2} + y^{2}\right)^{2}$$$ 关于$$$x$$$的积分
您的输入
求$$$\int \left(x^{2} + y^{2}\right)^{2}\, dx$$$。
解答
Expand the expression:
$${\color{red}{\int{\left(x^{2} + y^{2}\right)^{2} d x}}} = {\color{red}{\int{\left(x^{4} + 2 x^{2} y^{2} + y^{4}\right)d x}}}$$
逐项积分：
$${\color{red}{\int{\left(x^{4} + 2 x^{2} y^{2} + y^{4}\right)d x}}} = {\color{red}{\left(\int{x^{4} d x} + \int{y^{4} d x} + \int{2 x^{2} y^{2} d x}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=4$$$：
$$\int{y^{4} d x} + \int{2 x^{2} y^{2} d x} + {\color{red}{\int{x^{4} d x}}}=\int{y^{4} d x} + \int{2 x^{2} y^{2} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=\int{y^{4} d x} + \int{2 x^{2} y^{2} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$
应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=y^{4}$$$：
$$\frac{x^{5}}{5} + \int{2 x^{2} y^{2} d x} + {\color{red}{\int{y^{4} d x}}} = \frac{x^{5}}{5} + \int{2 x^{2} y^{2} d x} + {\color{red}{x y^{4}}}$$
对 $$$c=2 y^{2}$$$ 和 $$$f{\left(x \right)} = x^{2}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$\frac{x^{5}}{5} + x y^{4} + {\color{red}{\int{2 x^{2} y^{2} d x}}} = \frac{x^{5}}{5} + x y^{4} + {\color{red}{\left(2 y^{2} \int{x^{2} d x}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=2$$$：
$$\frac{x^{5}}{5} + x y^{4} + 2 y^{2} {\color{red}{\int{x^{2} d x}}}=\frac{x^{5}}{5} + x y^{4} + 2 y^{2} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{5}}{5} + x y^{4} + 2 y^{2} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
因此，
$$\int{\left(x^{2} + y^{2}\right)^{2} d x} = \frac{x^{5}}{5} + \frac{2 x^{3} y^{2}}{3} + x y^{4}$$
化简：
$$\int{\left(x^{2} + y^{2}\right)^{2} d x} = x \left(\frac{x^{4}}{5} + \frac{2 x^{2} y^{2}}{3} + y^{4}\right)$$
加上积分常数：
$$\int{\left(x^{2} + y^{2}\right)^{2} d x} = x \left(\frac{x^{4}}{5} + \frac{2 x^{2} y^{2}}{3} + y^{4}\right)+C$$
答案
$$$\int \left(x^{2} + y^{2}\right)^{2}\, dx = x \left(\frac{x^{4}}{5} + \frac{2 x^{2} y^{2}}{3} + y^{4}\right) + C$$$A