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

Problem

(∫_^)(x*cos(5*x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function x and a trigonometric function cos(5*x) use integration by parts.

2. Assign variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x and d(v)=cos(5*x)*d(x)

3. Differentiate u to find d(u)

d(u)=d(x)

4. Integrate d(v) to find v

v=(∫_^)(cos(5*x)*d(x))=sin(5*x)/5

5. Apply the integration by parts formula.

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

6. Evaluate the remaining integral.

(∫_^)(sin(5*x)/5*d(x))=1/5*(∫_^)(sin(5*x)*d(x))=1/5⋅(−cos(5*x)/5)=−cos(5*x)/25

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

(x*sin(5*x))/5−(−cos(5*x)/25)+C

Final Answer

(∫_^)(x*cos(5*x)*d(x))=(x*sin(5*x))/5+cos(5*x)/25+C

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