I did the first one for you, not gonna do all of em, but hopefully you are looking for explain.
First, split it into five separate integrals (one for each additive term)
8x^3 becomes 2x^4 (integrate by raising the exponent by 1 and then dividing the coefficient by the new exponent).
2/x becomes 2ln(x) (not going to prove it, but the derivative of ln(x) = 1/x is a rule, constant 2 just stays there)
-5e^x becomes -5e^x (the integral of e^x = e^x, the chain rule acting on e^x only produces the identity factor and -5 remains as the constant coefficient)
16(x-7)^(1/3) becomes 12(x-7)^(4/3). (Again, raise the exponent by 1, then divide by the new exponent. The chain rule acting on (x-7) only produces a factor of 1, again the multiplicative identity)
-7 becomes -7x. (Integrating a constant just requires you to slap an x onto it)
Thus your answer is 2x^4 +2ln(x) -5e^x + 12(x-7)^(4/3) -7x + C. Since you're doing indefinite integrals, don't forget the +C at the end.
I split it up, but a constant is a constant, so I combined all 5 constants into one.