Evaluate the Integral integral of (e^t+1)^2 with respect to t

Problem

(∫_^)((et+1)2*d(t))

Solution

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

(et+1)2=(et)2+2*(et)*(1)+1

2. Simplify the terms using exponent rules, specifically (et)2=e(2*t)

(et+1)2=e(2*t)+2*et+1

3. Apply the sum rule for integration to integrate each term individually.

(∫_^)((e(2*t)+2*et+1)*d(t))=(∫_^)(e(2*t)*d(t))+(∫_^)(2*et*d(t))+(∫_^)(1*d(t))

4. Evaluate the integrals using the power rule and the exponential rule (∫_^)(e(a*t)*d(t))=1/a*e(a*t)

(∫_^)(e(2*t)*d(t))=1/2*e(2*t)

(∫_^)(2*et*d(t))=2*et

(∫_^)(1*d(t))=t

5. Combine the results and add the constant of integration C

1/2*e(2*t)+2*et+t+C

Final Answer

(∫_^)((et+1)2*d(t))=1/2*e(2*t)+2*et+t+C

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