Evaluate the Integral integral of 5xcos(2x) with respect to x

Problem

(∫_^)(5*x*cos(2*x)*d(x))

Solution

1. Identify the method of integration by parts, which is defined by the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

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

3. Differentiate u to find d(u)=5*d(x) and integrate d(v) to find v=1/2*sin(2*x)

4. Substitute these components into the integration by parts formula.

(∫_^)(5*x*cos(2*x)*d(x))=(5*x)*(1/2*sin(2*x))−(∫_^)(1/2*sin(2*x)*(5)*d(x))

5. Simplify the expression before performing the second integration.

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

6. Evaluate the remaining integral using the rule (∫_^)(sin(a*x)*d(x))=−1/a*cos(a*x)

(∫_^)(sin(2*x)*d(x))=−1/2*cos(2*x)

7. Combine the results and add the constant of integration C

(∫_^)(5*x*cos(2*x)*d(x))=(5*x)/2*sin(2*x)−5/2*(−1/2*cos(2*x))+C

8. Simplify the final coefficients.

(∫_^)(5*x*cos(2*x)*d(x))=(5*x)/2*sin(2*x)+5/4*cos(2*x)+C

Final Answer

(∫_^)(5*x*cos(2*x)*d(x))=(5*x)/2*sin(2*x)+5/4*cos(2*x)+C

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