The coefficient 4 is the derivative of the exponent 4x:
Let u = 4x. Then the derivative of u with respect to x is 4. In symbols,
Thinking of as a fraction and multiplying both sides by dx, we can write
du = 4dx.
Now we can rewrite the integral using u instead of x.
We know that
Putting the original variable back in, we have
e4x + C.
We conclude that