Integral of $$$e^{x^{2}} - \sin{\left(x \right)}$$$

Find $$$\int \left(e^{x^{2}} - \sin{\left(x \right)}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(e^{x^{2}} - \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{e^{x^{2}} d x} - \int{\sin{\left(x \right)} d x}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

This integral (Imaginary Error Function) does not have a closed form:

$$\cos{\left(x \right)} + {\color{red}{\int{e^{x^{2}} d x}}} = \cos{\left(x \right)} + {\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2}\right)}}$$

Therefore,

$$\int{\left(e^{x^{2}} - \sin{\left(x \right)}\right)d x} = \cos{\left(x \right)} + \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2}$$

Add the constant of integration:

$$\int{\left(e^{x^{2}} - \sin{\left(x \right)}\right)d x} = \cos{\left(x \right)} + \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2}+C$$

$$$\int \left(e^{x^{2}} - \sin{\left(x \right)}\right)\, dx = \left(\cos{\left(x \right)} + \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2}\right) + C$$$A