Evaluate the Integral integral of sin(2t) with respect to t

Problem

(∫_^)(sin(2*t)*d(t))

Solution

1. Identify the integral as a basic trigonometric integral requiring a usubstitution or the reverse chain rule for a linear argument.

2. Apply the substitution u=2*t which implies d(u)=2*d(t) or d(t)=1/2*d(u)

3. Substitute the variables into the integral to get (∫_^)(sin(u)⋅1/2*d(u))

4. Integrate the sine function using the rule (∫_^)(sin(u)*d(u))=−cos(u)+C

5. Simplify the expression by multiplying the constants to get −1/2*cos(u)+C

6. Back-substitute u=2*t to return to the original variable.

Final Answer

(∫_^)(sin(2*t)*d(t))=−1/2*cos(2*t)+C

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