First, we want to write the function in a form that will be easier to integrate. Any radical terms will be turned into rational exponents.
f(x) = ∫(3x5 + 15xx1/2 - 48e-2x + (48 / (1 + x2)))dx
f(x) = ∫(3x5 + 15x3/2 - 48e-2x + (48 / (1 + x2))dx
Overall, you will use the power rule for integrals to evaluate the integral. I think you easily integrate the first two terms. The last two terms require a different approach. In the last two terms, you will need to use substitution. Let call these last two terms sub-integrals.
Let u = -2x
du = -2dx
(-1/2)du = dx
Plug in in these variables to get an integral
For 48∫(1 / (1 + x2)))dx
The anti-derivative of 1/(1+x2) is tan-1(x).
48∫(1 / (1 + x2)))dx = 48tan-1(x)
Then you are going to take the sum of these integrals. That will be your final integral. Make sure to add +C at the end.