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

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

Integrate term by term:

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

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

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

The integral of the exponential function is $$$\int{e^{x} d x} = e^{x}$$$:

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

Therefore,

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

Add the constant of integration:

$$\int{\left(e^{x} - \sin{\left(x \right)}\right)d x} = e^{x} + \cos{\left(x \right)}+C$$

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