∫e^x dx
The integral of e^x dx is given by ∫e^x dx = e^x + C, where C is the constant of integration
The integral of e^x dx is given by ∫e^x dx = e^x + C, where C is the constant of integration.
To derive this result, let’s consider the function f(x) = e^x. We want to find its antiderivative or indefinite integral.
One way to approach this is to use the basic rule for differentiation: the derivative of e^x is e^x itself. This suggests that the antiderivative of e^x should be of the form e^x.
We can test this assumption by taking the derivative of e^x:
d/dx (e^x) = e^x
Indeed, the derivative of e^x matches the original function, confirming our assumption.
Therefore, the indefinite integral of e^x is e^x + C, where C is an arbitrary constant. This means that for any value of C, the function F(x) = e^x + C is an antiderivative of e^x.
To understand the constant of integration, let’s consider the derivative of F(x) = e^x + C with respect to x.
d/dx (e^x + C) = d/dx (e^x) + d/dx (C) = e^x + 0 = e^x.
Note that the derivative of a constant term, C in this case, is always zero.
Therefore, although different constant values will yield different specific antiderivatives, they will all have the same derivative, e^x.
In conclusion, the indefinite integral of e^x is e^x + C, where C is an arbitrary constant.
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