How to Integrate the Function ∫a^x dx: Step-by-Step Derivation and Power Rule of Integration

∫a^x dx

To integrate the function ∫a^x dx, where a is a constant, we can use the power rule of integration

To integrate the function ∫a^x dx, where a is a constant, we can use the power rule of integration. The power rule states that when integrating a function of the form x^n, where n is any real number except -1, we can evaluate it as (1/(n+1)) * x^(n+1).

Using this rule, let’s integrate the function ∫a^x dx:
∫a^x dx = (1/ln(a)) * a^x + C

where ln(a) represents the natural logarithm of a, and C is the constant of integration.

This result can be obtained by letting u = a^x, and then applying the chain rule for integration, which states that ∫f'(u)*du = f(u) + C.

Now, let’s do a step-by-step derivation to justify this result:

1. Start by deriving the function u = a^x with respect to x.
du/dx = (ln(a))*a^x

2. Rearrange the equation to solve for dx:
dx = (1/ln(a)) * a^x * du

3. Substitute dx from step 2 and u from step 1 into the integral:
∫a^x dx = ∫(1/ln(a)) * a^x * du

4. Simplify the expression:
∫(1/ln(a)) * a^x * du = (1/ln(a)) * ∫a^x du

5. Integrate ∫a^x du using the power rule:
(1/ln(a)) * ∫a^x du = (1/ln(a)) * (a^x + C)

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

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