Evaluate the Integral integral of e^(3x-1) with respect to x

Problem

(∫_^)(e(3*x−1)*d(x))

Solution

1. Identify the form of the integral, which is an exponential function with a linear exponent.

2. Apply the substitution method by letting u=3*x−1

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=3*d(x) or d(x)=1/3*d(u)

4. Substitute the variables into the integral to get (∫_^)(eu⋅1/3*d(u))

5. Factor out the constant 1/3 to get 1/3*(∫_^)(eu*d(u))

6. Integrate the expression using the rule (∫_^)(eu*d(u))=eu+C

7. Substitute back the original expression for u to obtain the final result.

Final Answer

(∫_^)(e(3*x−1)*d(x))=(e(3*x−1))/3+C

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