Evaluate the Integral

Problem

(∫_^)(e(3*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(3*x)*d(x)

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

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

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

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

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

5. Substitute this result back into the equation from step 3.

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

6. Distribute the constant 2/3 through the parentheses.

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

7. Add 4/9*(∫_^)(e(3*x)*cos(2*x)*d(x)) to both sides to group the integral terms.

13/9*(∫_^)(e(3*x)*cos(2*x)*d(x))=1/3*e(3*x)*cos(2*x)+2/9*e(3*x)*sin(2*x)

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

(∫_^)(e(3*x)*cos(2*x)*d(x))=9/13*(1/3*e(3*x)*cos(2*x)+2/9*e(3*x)*sin(2*x))+C

9. Simplify the final expression.

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

Final Answer

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

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