Evaluate the Integral integral of a^x with respect to x

Problem

(∫_^)(ax*d(x))

Solution

1. Identify the integrand as an exponential function with a constant base a where a>0 and a≠1

2. Rewrite the expression using the natural base e and the identity ax=e(x*ln(a))

3. Apply the integration rule for exponential functions of the form e(k*x) which states (∫_^)(e(k*x)*d(x))=(e(k*x))/k+C

4. Substitute k=ln(a) into the formula to obtain (e(x*ln(a)))/ln(a)+C

5. Simplify the numerator back to ax and include the constant of integration C

Final Answer

(∫_^)(ax*d(x))=(ax)/ln(a)+C

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