Using the Power Rule | Evaluating the Indefinite Integral of a^u

∫(a^u)du

The integral you’ve provided, ∫(a^u)du, represents the indefinite integral of the function f(u) = a^u with respect to u

The integral you’ve provided, ∫(a^u)du, represents the indefinite integral of the function f(u) = a^u with respect to u. To evaluate this integral, we can use the power rule of integration.

The power rule states that if we have a function of the form f(x) = x^n, where n is any real number except -1, then the indefinite integral of f(x) with respect to x is given by F(x) = (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

Applying the power rule to the integral ∫(a^u)du, we can rewrite a^u as e^(ln(a^u)) to make use of the properties of exponents and logarithms. Then, we have ∫(a^u)du = ∫(e^(ln(a^u)))du.

Now, let’s focus on the term inside the integral, e^(ln(a^u)). Since e^x is the inverse function of ln(x), we can rewrite this expression further as e^(ln(a^u)) = a^u.

Now our integral becomes ∫(a^u)du = ∫a^udu.

Using the power rule, where the base a is constant, we can integrate a^u as (1/ln(a)) * a^u + C, where C is the constant of integration.

Therefore, the final result is ∫(a^u)du = (1/ln(a)) * a^u + C, where C is the constant of integration.

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