Evaluate the Integral integral of e^(5x)(5) with respect to x

Problem

(∫_^)(e(5)(5*x)*d(x))

Solution

1. Identify the structure of the integrand, which is in the form ƒ*(g(x))⋅g(x)′

2. Apply the substitution method by letting u=5*x

3. Calculate the differential d(u)=5*d(x)

4. Substitute the values into the integral to get (∫_^)(eu*d(u))

5. Integrate the exponential function using the rule (∫_^)(eu*d(u))=eu+C

6. Back-substitute u=5*x to return to the original variable.

Final Answer

(∫_^)(e(5)(5*x)*d(x))=e(5*x)+C

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