Integral of $$$e^{\frac{t}{2}} - \frac{5}{t^{2}}$$$ with respect to $$$x$$$

Find $$$\int \left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)\, dx$$$.

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=e^{\frac{t}{2}} - \frac{5}{t^{2}}$$$:

$${\color{red}{\int{\left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)d x}}} = {\color{red}{x \left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)}}$$

Therefore,

$$\int{\left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)d x} = x \left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)$$

Simplify:

$$\int{\left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)d x} = x e^{\frac{t}{2}} - \frac{5 x}{t^{2}}$$

Add the constant of integration:

$$\int{\left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)d x} = x e^{\frac{t}{2}} - \frac{5 x}{t^{2}}+C$$

$$$\int \left(e^{\frac{t}{2}} - \frac{5}{t^{2}}\right)\, dx = \left(x e^{\frac{t}{2}} - \frac{5 x}{t^{2}}\right) + C$$$A