Evaluate the Integral

Problem

(∫_0^4*π)(t2*sin(2*t)*d(t))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=t2 and d(v)=sin(2*t)*d(t)

2. Differentiate u to find d(u)=2*t*d(t) and integrate d(v) to find v=−1/2*cos(2*t)

3. Apply the integration by parts formula for the first time:

(∫_^)(t2*sin(2*t)*d(t))=−(t2)/2*cos(2*t)−(∫_^)(−t*cos(2*t)*d(t))

4. Simplify the expression:

(∫_^)(t2*sin(2*t)*d(t))=−(t2)/2*cos(2*t)+(∫_^)(t*cos(2*t)*d(t))

5. Apply integration by parts again for the remaining integral (∫_^)(t*cos(2*t)*d(t)) Let u=t and d(v)=cos(2*t)*d(t) so d(u)=d(t) and v=1/2*sin(2*t)

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

6. Integrate the final term:

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

7. Combine all parts to find the general antiderivative:

(∫_^)(t2*sin(2*t)*d(t))=−(t2)/2*cos(2*t)+t/2*sin(2*t)+1/4*cos(2*t)

8. Evaluate the definite integral from 0 to 4*π by substituting the upper and lower limits:

[−(t2)/2*cos(2*t)+t/2*sin(2*t)+1/4*cos(2*t)](4*π)0

9. Substitute 4*π

(−((4*π)2)/2*cos(8*π)+(4*π)/2*sin(8*π)+1/4*cos(8*π))=−8*π2+0+1/4

10. Substitute 0

(−0/2*cos(0)+0/2*sin(0)+1/4*cos(0))=0+0+1/4

11. Subtract the lower limit value from the upper limit value:

−8*π2+1/4−1/4=−8*π2

Final Answer

(∫_0^4*π)(t2*sin(2*t)*d(t))=−8*π2

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