Evaluate the Integral

Problem

(∫_^)(5*(t2−5*t−2)*d(t))

Solution

1. Factor out the constant 5 from the integral to simplify the expression.

(∫_^)(5*(t2−5*t−2)*d(t))=5*(∫_^)((t2−5*t−2)*d(t))

2. Apply the sum rule for integration, which allows for the integration of each term individually.

5*(∫_^)((t2−5*t−2)*d(t))=5*((∫_^)(t2*d(t))−(∫_^)(5*t*d(t))−(∫_^)(2*d(t)))

3. Apply the power rule for integration, (∫_^)(tn*d(t))=(t(n+1))/(n+1) to each term.

5*((t3)/3−(5*t2)/2−2*t)+C

4. Distribute the constant 5 to each term in the expression.

(5*t3)/3−(25*t2)/2−10*t+C

Final Answer

(∫_^)(5*(t2−5*t−2)*d(t))=(5*t3)/3−(25*t2)/2−10*t+C

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