Integralen av $$$\frac{i n t^{2} u}{2} + \cos{\left(x \right)}$$$ med avseende på $$$x$$$

Bestäm $$$\int \left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{\frac{i n t^{2} u}{2} d x} + \int{\cos{\left(x \right)} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=\frac{i n t^{2} u}{2}$$$:

$$\int{\cos{\left(x \right)} d x} + {\color{red}{\int{\frac{i n t^{2} u}{2} d x}}} = \int{\cos{\left(x \right)} d x} + {\color{red}{\left(\frac{i n t^{2} u x}{2}\right)}}$$

Integralen av cosinus är $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$\frac{i n t^{2} u x}{2} + {\color{red}{\int{\cos{\left(x \right)} d x}}} = \frac{i n t^{2} u x}{2} + {\color{red}{\sin{\left(x \right)}}}$$

Alltså,

$$\int{\left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)d x} = \frac{i n t^{2} u x}{2} + \sin{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)d x} = \frac{i n t^{2} u x}{2} + \sin{\left(x \right)}+C$$

$$$\int \left(\frac{i n t^{2} u}{2} + \cos{\left(x \right)}\right)\, dx = \left(\frac{i n t^{2} u x}{2} + \sin{\left(x \right)}\right) + C$$$A