$$$a^{2} \cos{\left(x \right)} - x^{2}$$$ 對 $$$x$$$ 的積分

求$$$\int \left(a^{2} \cos{\left(x \right)} - x^{2}\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(a^{2} \cos{\left(x \right)} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{a^{2} \cos{\left(x \right)} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$$\int{a^{2} \cos{\left(x \right)} d x} - {\color{red}{\int{x^{2} d x}}}=\int{a^{2} \cos{\left(x \right)} d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{a^{2} \cos{\left(x \right)} d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=a^{2}$$$ 與 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$：

$$- \frac{x^{3}}{3} + {\color{red}{\int{a^{2} \cos{\left(x \right)} d x}}} = - \frac{x^{3}}{3} + {\color{red}{a^{2} \int{\cos{\left(x \right)} d x}}}$$

餘弦函數的積分為 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$：

$$a^{2} {\color{red}{\int{\cos{\left(x \right)} d x}}} - \frac{x^{3}}{3} = a^{2} {\color{red}{\sin{\left(x \right)}}} - \frac{x^{3}}{3}$$

因此，

$$\int{\left(a^{2} \cos{\left(x \right)} - x^{2}\right)d x} = a^{2} \sin{\left(x \right)} - \frac{x^{3}}{3}$$

加上積分常數：

$$\int{\left(a^{2} \cos{\left(x \right)} - x^{2}\right)d x} = a^{2} \sin{\left(x \right)} - \frac{x^{3}}{3}+C$$

$$$\int \left(a^{2} \cos{\left(x \right)} - x^{2}\right)\, dx = \left(a^{2} \sin{\left(x \right)} - \frac{x^{3}}{3}\right) + C$$$A