(a^x/lna) + C
∫a^x dx
The given expression is the antiderivative or indefinite integral of the function f(x) = a^x with respect to x, where a is a constant and C is the constant of integration.
To find this antiderivative, we can use the power rule of integration, which states that if f(x) = x^n, then its antiderivative F(x) = (x^(n+1))/(n+1) + C.
Using this rule, we can rewrite the given function as f(x) = a^x = (e^(ln(a))^x), since a^x = e^(ln(a^x)) = e^(x*ln(a)).
Let n = ln(a), then f(x) = e^(n*x), and the antiderivative of f(x) with respect to x is therefore F(x) = (e^(n*x))/(n) + C.
Substituting back in n = ln(a), we get F(x) = (a^x)/ln(a) + C, which is the given expression.
Therefore, the antiderivative of a^x with respect to x is (a^x)/ln(a) + C.
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