`int(csc^2(t)-2e^t)dt`

Apply the sum rule,

`=intcsc^2(t)dt-int(2e^tdt`

We now the following common integrals,

`intcsc^2(x)=-cot(x)` and `inte^xdx=e^x`

evaluate using the above,

`=-cot(t)-2e^t+C` , C is constant

`int(csc^2(t)-2e^t)dt`

Apply the sum rule,

`=intcsc^2(t)dt-int(2e^tdt`

We now the following common integrals,

`intcsc^2(x)=-cot(x)` and `inte^xdx=e^x`

evaluate using the above,

`=-cot(t)-2e^t+C` , C is constant