Exploring the Indefinite Integral of a^x | A Guide with Step-by-Step Procedure and Special Cases

∫ a^x dx

The integral of a^x dx, where a is a positive constant, represents the indefinite integral or antiderivative of the function a^x with respect to x

The integral of a^x dx, where a is a positive constant, represents the indefinite integral or antiderivative of the function a^x with respect to x. To solve this integral, we need to use a specific technique called the exponential function.

To start, let’s assume that a is not equal to 1 (a≠1). The integral of a^x dx can be solved as follows:

∫ a^x dx = (1/ln(a)) * a^x + C

where C is the constant of integration.

To understand the above formula, let’s go through the steps:

Step 1: Differentiate a^x to find the derivative. We have d/dx (a^x) = a^x * ln(a).
Step 2: Now, we want to find the integral of a^x. By inverting the derivative, we obtain ∫ a^x dx = (1/ln(a)) * a^x + C.

Note: If a = 1, then the integral becomes a special case, as 1^x equals 1 for all values of x. In this case, the integral simplifies to:

∫ dx = x + C

So, the result depends on the value of a. If a≠1, the resulting indefinite integral of a^x dx is (1/ln(a)) * a^x + C. If a=1, then the indefinite integral is simply x + C.

Remember that the antiderivative of a function gives us a family of functions that differ by a constant value (C).

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