Solving the Indefinite Integral of a^x Using the Power Rule

int a^x dx

The expression “int a^x dx” represents the indefinite integral of the function f(x) = a^x with respect to x

The expression “int a^x dx” represents the indefinite integral of the function f(x) = a^x with respect to x. To solve this integral, we can use the power rule of integration.

The power rule states that if we have an integral of the form ∫x^n dx, where n is a constant, then the antiderivative is given by (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

Applying the power rule to the integral of a^x, we consider a as a constant and treat x as the variable. Thus, the integral becomes:

∫a^x dx = (1/(ln(a))) * a^x + C,

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

Note that the natural logarithm function, ln(x), is the inverse of the exponential function e^x. Therefore, ln(a) represents the exponent to which e must be raised to obtain a.

Thus, the final answer for the integral of a^x dx is (1/(ln(a))) * a^x + C.

More Answers:

A Comprehensive Guide to Integrating Powers: Power Rule and Special Cases
The Magic of Logs: Evaluating the Integral of du/u with Natural Logarithms
The Integral of e^x: Calculating the Area Under the Curve with the Power Rule for Integration

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