Evaluate the Integral integral of te^(-9t) with respect to t

Problem

(∫_^)(t*e(−9*t)*d(t))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function t and an exponential function e(−9*t) use integration by parts.

2. Choose the parts for the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=t and d(v)=e(−9*t)*d(t)

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

d(u)=d(t)

v=(∫_^)(e(−9*t)*d(t))=−1/9*e(−9*t)

4. Apply the integration by parts formula.

(∫_^)(t*e(−9*t)*d(t))=t*(−1/9*e(−9*t))−(∫_^)((−1/9*e(−9*t))*d(t))

5. Simplify the expression and evaluate the remaining integral.

(∫_^)(t*e(−9*t)*d(t))=−t/9*e(−9*t)+1/9*(∫_^)(e(−9*t)*d(t))

(∫_^)(t*e(−9*t)*d(t))=−t/9*e(−9*t)+1/9*(−1/9*e(−9*t))+C

6. Factor out common terms to simplify the final result.

(∫_^)(t*e(−9*t)*d(t))=−t/9*e(−9*t)−1/81*e(−9*t)+C

Final Answer

(∫_^)(t*e(−9*t)*d(t))=−1/9*t*e(−9*t)−1/81*e(−9*t)+C

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