Evaluate the Integral

Problem

(∫_^)(((3+e^x)^2)/(e^x)*d(x))

Solution

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

(∫_^)((9+6*e^x+e^(2*x))/(e^x)*d(x))

2. Divide each term in the numerator by the denominator e^x to simplify the integrand.

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

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

(∫_^)((9*e^(−x)+6+e^x)*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+e^x+C

Final Answer

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

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