A Guide to Finding the Antiderivative of a^x: Step-by-Step Explanation and Formula

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.

More Answers:

Derivative of e^x: Understanding the Fundamental Result and Calculation Methods in Calculus
Math Made Easy: Understanding the Antiderivative of e^x using the Power Rule
The Derivative of f(x) = a^x using Logarithmic Differentiation: Simplified Step-by-Step Guide

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