Evaluate the Integral integral of te^t with respect to t

Problem

(∫_^)(t*et*d(t))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function t and an exponential function et use integration by parts.

2. Assign variables for the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=t and d(v)=et*d(t)

3. Differentiate u to find d(u)=d(t) and integrate d(v) to find v=et

4. Apply the formula by substituting u v d(u) and d(v) into the integration by parts equation.

(∫_^)(t*et*d(t))=t*et−(∫_^)(et*d(t))

5. Evaluate the remaining integral (∫_^)(et*d(t)) which results in et

6. Combine the terms and add the constant of integration C

t*et−et+C

Final Answer

(∫_^)(t*et*d(t))=t*et−et+C

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