Evaluate the Integral

Problem

(∫_^)(e(−x)*cos(2*x)*d(x))

Solution

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

2. Apply the formula for the first time:

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

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

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

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

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

4. Substitute this result back into the original equation:

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

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

5. Solve for the integral by adding 4*(∫_^)(e(−x)*cos(2*x)*d(x)) to both sides:

5*(∫_^)(e(−x)*cos(2*x)*d(x))=−e(−x)*cos(2*x)+2*e(−x)*sin(2*x)

6. Divide by 5 and add the constant of integration C

(∫_^)(e(−x)*cos(2*x)*d(x))=(−e(−x)*cos(2*x)+2*e(−x)*sin(2*x))/5+C

Final Answer

(∫_^)(e(−x)*cos(2*x)*d(x))=(e(−x)*(2*sin(2*x)−cos(2*x)))/5+C

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