Evaluate the Integral

Problem

(∫_^)(((3+ex)2)/(ex)*d(x))

Solution

1. Expand the numerator using the square of a binomial formula (a+b)2=a2+2*a*b+b2

(∫_^)((9+6*ex+e(2*x))/(ex)*d(x))

2. Divide each term in the numerator by the denominator ex to simplify the integrand.

(∫_^)((9/(ex)+(6*ex)/(ex)+(e(2*x))/(ex))*d(x))

3. Rewrite the expression using negative exponents and simplify the fractions.

(∫_^)((9*e(−x)+6+ex)*d(x))

4. Integrate each term individually using the power rule and the rule for exponential functions (∫_^)(e(a*x)*d(x))=1/a*e(a*x)

−9*e(−x)+6*x+ex+C

Final Answer

(∫_^)(((3+ex)2)/(ex)*d(x))=−9*e(−x)+6*x+ex+C

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