Learn how to find antiderivatives of e^x and understand the power rule for integration

e^x + C

∫e^x dx

The expression e^x + C represents the antiderivative or indefinite integral of the function f(x) = e^x, where C is the constant of integration. The notation e^x is shorthand for the exponential function, which is defined as f(x) = e^x = exp(x) = lim[n→∞]((1 + x/n)^n), where e is the mathematical constant approximately equal to 2.71828.

To find the antiderivative of f(x) = e^x, we can use the power rule for integration, which states that the antiderivative of x^n is (1/(n+1))x^(n+1) + C, where C is the constant of integration. Using this rule, we have:

∫e^x dx = e^x + C

This is because the derivative of e^x is e^x, which means that e^x + C is a function whose derivative is e^x. The addition of the constant C is required because the derivative of a constant is zero, so any constant added to the function will not affect its derivative.

Therefore, e^x + C is one of the infinitely many antiderivatives of e^x. This expression represents a family of functions that differ only in the value of the constant C. To find a specific antiderivative of e^x, we need to know the value of the constant C, which can be determined by evaluating the function at a particular point or using initial conditions.

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