Evaluate the Integral integral of 5xcos(4x) with respect to x

Problem

(∫_^)(5*x*cos(4*x)*d(x))

Solution

1. Identify the integration method as integration by parts, using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign variables for the parts, choosing u=5*x and d(v)=cos(4*x)*d(x)

3. Differentiate u to find d(u)=5*d(x)

4. Integrate d(v) to find v=1/4*sin(4*x)

5. Substitute these values into the integration by parts formula:

(∫_^)(5*x*cos(4*x)*d(x))=(5*x)*(1/4*sin(4*x))−(∫_^)(1/4*sin(4*x)*(5*d(x)))

6. Simplify the expression before performing the final integration:

(∫_^)(5*x*cos(4*x)*d(x))=(5*x)/4*sin(4*x)−5/4*(∫_^)(sin(4*x)*d(x))

7. Evaluate the remaining integral:

(∫_^)(sin(4*x)*d(x))=−1/4*cos(4*x)

8. Combine the terms and add the constant of integration C

(∫_^)(5*x*cos(4*x)*d(x))=(5*x)/4*sin(4*x)−5/4*(−1/4*cos(4*x))+C

9. Simplify the final coefficients:

(∫_^)(5*x*cos(4*x)*d(x))=(5*x)/4*sin(4*x)+5/16*cos(4*x)+C

Final Answer

(∫_^)(5*x*cos(4*x)*d(x))=(5*x)/4*sin(4*x)+5/16*cos(4*x)+C

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