Evaluate the Integral

Problem

(∫_^)(e(2*x)*cos(3*x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=cos(3*x) and d(v)=e(2*x)*d(x)

2. Calculate the differentials d(u)=−3*sin(3*x)*d(x) and v=1/2*e(2*x)

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

(∫_^)(e(2*x)*cos(3*x)*d(x))=1/2*e(2*x)*cos(3*x)−(∫_^)(1/2*e(2*x)*(−3*sin(3*x))*d(x))

4. Simplify the expression to prepare for a second round of integration by parts.

(∫_^)(e(2*x)*cos(3*x)*d(x))=1/2*e(2*x)*cos(3*x)+3/2*(∫_^)(e(2*x)*sin(3*x)*d(x))

5. Apply integration by parts again to the new integral (∫_^)(e(2*x)*sin(3*x)*d(x)) Let u=sin(3*x) and d(v)=e(2*x)*d(x) so d(u)=3*cos(3*x)*d(x) and v=1/2*e(2*x)

(∫_^)(e(2*x)*sin(3*x)*d(x))=1/2*e(2*x)*sin(3*x)−(∫_^)(3/2*e(2*x)*cos(3*x)*d(x))

6. Substitute this result back into the main equation.

(∫_^)(e(2*x)*cos(3*x)*d(x))=1/2*e(2*x)*cos(3*x)+3/2*(1/2*e(2*x)*sin(3*x)−3/2*(∫_^)(e(2*x)*cos(3*x)*d(x)))

7. Distribute the constant and group the integral terms on one side.

(∫_^)(e(2*x)*cos(3*x)*d(x))=1/2*e(2*x)*cos(3*x)+3/4*e(2*x)*sin(3*x)−9/4*(∫_^)(e(2*x)*cos(3*x)*d(x))

8. Add 9/4*(∫_^)(e(2*x)*cos(3*x)*d(x)) to both sides.

13/4*(∫_^)(e(2*x)*cos(3*x)*d(x))=1/2*e(2*x)*cos(3*x)+3/4*e(2*x)*sin(3*x)

9. Solve for the integral by multiplying by 4/13 and adding the constant of integration C

(∫_^)(e(2*x)*cos(3*x)*d(x))=4/13*(1/2*e(2*x)*cos(3*x)+3/4*e(2*x)*sin(3*x))+C

Final Answer

(∫_^)(e(2*x)*cos(3*x)*d(x))=(2*e(2*x)*cos(3*x))/13+(3*e(2*x)*sin(3*x))/13+C

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