Evaluate the integral of 2 raised to the power of 3 times x, dx.

Question

    None of these

Answer 1

As a reminder, the derivative rule for an exponential goes like this:

When you to evaluate ∫2^3x, you are asking, "What expression, when we take it's derivative, gives us 2^3x?

Let's first try 2^3x. Take its derivative using the rule above PLUS the chain rule and we get:

ln(2)•2^3x•(3) or 3ln(2)•2^3x which is close, but it has that extra "3ln(2)" on it. Can you see that if we use 2^3x/(3ln(2)) that the denominator would cancel out that extra term we got, and we would be left with just 2^3x (which is what we want)? Of course, we would need to add "+ C" to cover any constant term.