The antiderivative of e^ln(4)x is simply x.
To find the antiderivative of e^ln(4)x, you can use the substitution method by letting u = ln(4)x and du = 1/x dx. This will simplify the integral to just e^u, which has the antiderivative of e^u + C. Then, substitute back in u = ln(4)x to get the final answer of e^ln(4)x + C = 4x + C.
This is because e^ln(4)x is essentially the same as 4^x, which is the inverse function of log base 4. The antiderivative of log base 4 is simply x, so the antiderivative of e^ln(4)x must also be x.
Yes, you can also use the fact that e^ln(4)x is equivalent to 4^x, and use the power rule for integrals, which states that the antiderivative of x^n is (x^(n+1))/(n+1) + C. In this case, the antiderivative of 4^x would be (4^(x+1))/(x+1) + C, which simplifies to 4x + C.
Yes, the antiderivative of e^ln(4)x is always equal to x, as long as the base of the log function is the same as the base of the exponential function, as it is in this case. Otherwise, the antiderivative would be different.