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

Problem

(∫_^)(t*sin(2*t)*d(t))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function t and a trigonometric function sin(2*t) use integration by parts.

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

3. Differentiate u and integrate d(v)

d(u)=d(t)

v=(∫_^)(sin(2*t)*d(t))=−1/2*cos(2*t)

4. Apply the integration by parts formula.

(∫_^)(t*sin(2*t)*d(t))=t*(−1/2*cos(2*t))−(∫_^)((−1/2*cos(2*t))*d(t))

5. Simplify the expression and prepare for the second integral.

(∫_^)(t*sin(2*t)*d(t))=−t/2*cos(2*t)+1/2*(∫_^)(cos(2*t)*d(t))

6. Evaluate the remaining integral.

(∫_^)(cos(2*t)*d(t))=1/2*sin(2*t)

7. Combine all terms and add the constant of integration C

(∫_^)(t*sin(2*t)*d(t))=−t/2*cos(2*t)+1/2*(1/2*sin(2*t))+C

8. Simplify the final coefficients.

(∫_^)(t*sin(2*t)*d(t))=−t/2*cos(2*t)+1/4*sin(2*t)+C

Final Answer

(∫_^)(t*sin(2*t)*d(t))=−t/2*cos(2*t)+1/4*sin(2*t)+C

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