Evaluate the Integral

Problem

(∫_^)(e(3*x)*cos(4*x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=cos(4*x) and d(v)=e(3*x)*d(x)

2. Calculate the differentials d(u)=−4*sin(4*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(4*x)*d(x))=1/3*e(3*x)*cos(4*x)+4/3*(∫_^)(e(3*x)*sin(4*x)*d(x))

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

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

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

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

6. Distribute the constant and group the integral terms on the left side.

(∫_^)(e(3*x)*cos(4*x)*d(x))=1/3*e(3*x)*cos(4*x)+4/9*e(3*x)*sin(4*x)−16/9*(∫_^)(e(3*x)*cos(4*x)*d(x))

7. Add 16/9*(∫_^)(e(3*x)*cos(4*x)*d(x)) to both sides.

25/9*(∫_^)(e(3*x)*cos(4*x)*d(x))=1/3*e(3*x)*cos(4*x)+4/9*e(3*x)*sin(4*x)

8. Solve for the integral by multiplying by 9/25 and adding the constant of integration C

(∫_^)(e(3*x)*cos(4*x)*d(x))=9/25*(1/3*e(3*x)*cos(4*x)+4/9*e(3*x)*sin(4*x))+C

Final Answer

(∫_^)(e(3*x)*cos(4*x)*d(x))=(3*e(3*x)*cos(4*x)+4*e(3*x)*sin(4*x))/25+C

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