$$$\frac{i n t^{2} u}{2} + \cos{\left(x \right)}$$$ の $$$x$$$ に関する積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int \left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)\, dx$$$ を求めよ。
解答
項別に積分せよ:
$${\color{red}{\int{\left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{\frac{i n t^{2} u}{2} d x} + \int{\cos{\left(x \right)} d x}\right)}}$$
$$$c=\frac{i n t^{2} u}{2}$$$ に対して定数則 $$$\int c\, dx = c x$$$ を適用する:
$$\int{\cos{\left(x \right)} d x} + {\color{red}{\int{\frac{i n t^{2} u}{2} d x}}} = \int{\cos{\left(x \right)} d x} + {\color{red}{\left(\frac{i n t^{2} u x}{2}\right)}}$$
余弦の積分は$$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$\frac{i n t^{2} u x}{2} + {\color{red}{\int{\cos{\left(x \right)} d x}}} = \frac{i n t^{2} u x}{2} + {\color{red}{\sin{\left(x \right)}}}$$
したがって、
$$\int{\left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)d x} = \frac{i n t^{2} u x}{2} + \sin{\left(x \right)}$$
積分定数を加える:
$$\int{\left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)d x} = \frac{i n t^{2} u x}{2} + \sin{\left(x \right)}+C$$
解答
$$$\int \left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)\, dx = \left(\frac{i n t^{2} u x}{2} + \sin{\left(x \right)}\right) + C$$$A