Evaluate the Integral integral of e^(7x+4) with respect to x

Problem

(∫_^)(e(7*x+4)*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=7*x+4

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=7*d(x) or d(x)=1/7*d(u)

4. Substitute the expressions for u and d(x) into the integral to get (∫_^)(eu⋅1/7*d(u))

5. Factor out the constant 1/7 to get 1/7*(∫_^)(eu*d(u))

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

7. Back-substitute the original expression for u to return to the variable x

Final Answer

(∫_^)(e(7*x+4)*d(x))=(e(7*x+4))/7+C

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