Simplifying Integration | Finding the Antiderivative of e^u with Respect to u

∫e^u du

The integral of e^u with respect to u is given by ∫e^u du

The integral of e^u with respect to u is given by ∫e^u du. To find the antiderivative of e^u, we make use of the fact that the derivative of e^x with respect to x is also e^x.

In this case, we can rewrite the integral as:

∫e^u du = ∫(1)(e^u) du

Now, let’s apply the rule of integration known as u-substitution. We make a substitution by letting u be equal to a specific function within the integral. In this case, we choose u = u.

Differentiating both sides with respect to u, we get:

du = du

Now we can rewrite the integral in terms of the new variable:

∫e^u du = ∫(1)(e^u) du = ∫e^u du

Notice that the integrand matches the derivative of the new variable, du. When this happens, it simplifies the integration process and allows us to evaluate the integral more easily.

Therefore, the integral becomes:

∫e^u du = ∫ e^u du = e^u + C

where C is the constant of integration. Thus, the antiderivative of e^u with respect to u is e^u + C.

In conclusion, the integral of e^u with respect to u is e^u + C, where C is the constant of integration.

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