Find the Integral te^t

Problem

(∫_^)(t*et*d(t))

Solution

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

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

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

d(u)=d(t)

v=et

4. Apply the formula for integration by parts.

(∫_^)(t*et*d(t))=t*et−(∫_^)(et*d(t))

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

(∫_^)(et*d(t))=et

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

(∫_^)(t*et*d(t))=t*et−et+C

Final Answer

(∫_^)(t*et*d(t))=t*et−et+C

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