Evaluate the Integral

Problem

(∫_^)(((2+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

(∫_^)((4+4*ex+e(2*x))/(ex)*d(x))

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

(∫_^)((4/(ex)+(4*ex)/(ex)+(e(2*x))/(ex))*d(x))

3. Simplify the expressions using the laws of exponents.

(∫_^)((4*e(−x)+4+ex)*d(x))

4. Integrate each term individually using the power rule and the rule for exponential functions.

−4*e(−x)+4*x+ex+C

Final Answer

(∫_^)(((2+ex)2)/(ex)*d(x))=−4*e(−x)+4*x+ex+C

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