Integrate the following function: f(x) = 8x^3 + 1/x + 5

We can see that the function is a sum of three terms so we can deal with each term separately and add them up. The term 8x³ and 5 are relatively straightforward and follow the standard rules for integration: "raise the power by 1 and divide by the new power". Therefore 8x^3 becomes 8/4 x⁴ = 2x⁴ and the 5 becomes 5x. Then we look at the 1/x term. This is slightly more complicated as it we cannot follow that rule since, remembering 1/x is the same as x^-1, this would give us x⁰/0 which can't be true. Instead, we know that 1/x integrates to ln(x) (the natural logarithm). Finally, as with all indefinite integratals (integration without limits) we have to add a constant. The final answer is therefore 2x⁴ + ln(x) + 5x + c