Antiderivative of a^x
To find the antiderivative of a^x, where a is a constant, we need to evaluate the integral
To find the antiderivative of a^x, where a is a constant, we need to evaluate the integral. The antiderivative of a^x is denoted as ∫a^x dx.
To understand how to find the antiderivative, it is helpful to recall the formula for the antiderivative of a basic function. For any real number n (except -1), the antiderivative of x^n is (1/(n+1))x^(n+1) + C, where C is the constant of integration.
Using this formula, we can proceed to find the antiderivative of a^x. Let’s consider the case where a is not equal to 1.
∫a^x dx = 1/(ln(a)) * ∫(ln(a) * a^x) dx
Now, let’s focus on the integral of (ln(a) * a^x).
The integral of (ln(a) * a^x) dx can be evaluated using u-substitution. Let u = a^x and du = (ln(a) * a^x) dx.
So, ∫(ln(a) * a^x) dx = ∫du = u + C
= a^x + C
Reintroducing the 1/(ln(a)) factor outside the integral, we have:
∫a^x dx = 1/(ln(a)) * ∫(ln(a) * a^x) dx
= 1/(ln(a)) * (a^x + C)
= (1/(ln(a))) * a^x + C
Therefore, the antiderivative of a^x is given by (1/(ln(a))) * a^x + C, where C is the constant of integration.
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