Evaluate the Integral

Problem

(∫_^)(((6+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

(6+ex)2=36+12*ex+e(2*x)

2. Rewrite the integral by dividing each term in the numerator by the denominator ex

(∫_^)((36+12*ex+e(2*x))/(ex)*d(x))

3. Simplify the expression inside the integral using the properties of exponents.

(∫_^)((36*e(−x)+12+ex)*d(x))

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

−36*e(−x)+12*x+ex+C

Final Answer

(∫_^)(((6+ex)2)/(ex)*d(x))=−36*e(−x)+12*x+ex+C

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