Is this equation correct? and valid? $e^x=\frac{d e^x}{dx}=\int dx e^x = e^x$

Question

I just wanted to know whether the series of equations $e^x=\frac{d}{dx} e^x =\int e^x dx= e^x = ...$ is valid and well defined.

Answer 1

No, the equation is not correct. This is because the indefinite integral of a function is the antiderivative plus a constant, C. The constant C can be anything. For example if C=1 this equality will not hold.

However if you were to ask if the derivative is equal to the function, then this would be correct. In fact this holds for any nth derivative where n is an integer and n>0. $$e^x=\frac{d^n}{dx^n}(e^x)$$ In addition,$$\int^{x}_{-\infty}{e^t\ dt}=e^x$$ The improper integral on the left is also is equal to e^x.

You can also iterate the integral and that will also be equal. You can either use nested integrals or use Cauchy's Repeated Integral Formula. This formula is a little more advanced and may not be worth your time.