Evaluate the Integral

Problem

(∫_^)(e(2*x)*sin(2*x)*d(x))

Solution

1. Identify the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=sin(2*x) and d(v)=e(2*x)*d(x)

2. Calculate the differentials d(u)=2*cos(2*x)*d(x) and v=1/2*e(2*x)

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

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

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

4. Apply integration by parts again to the new integral (∫_^)(e(2*x)*cos(2*x)*d(x)) Let u=cos(2*x) and d(v)=e(2*x)*d(x)

5. Calculate the differentials d(u)=−2*sin(2*x)*d(x) and v=1/2*e(2*x)

6. Substitute these into the second integral.

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

(∫_^)(e(2*x)*cos(2*x)*d(x))=1/2*e(2*x)*cos(2*x)+(∫_^)(e(2*x)*sin(2*x)*d(x))

7. Combine the results back into the original equation. Let I=(∫_^)(e(2*x)*sin(2*x)*d(x))

I=1/2*e(2*x)*sin(2*x)−(1/2*e(2*x)*cos(2*x)+I)

I=1/2*e(2*x)*sin(2*x)−1/2*e(2*x)*cos(2*x)−I

8. Solve for I by adding I to both sides and dividing by 2.

2*I=1/2*e(2*x)*sin(2*x)−1/2*e(2*x)*cos(2*x)

I=1/4*e(2*x)*sin(2*x)−1/4*e(2*x)*cos(2*x)+C

Final Answer

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

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