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

Problem

(∫_^)(x*cos(4*x)*d(x))

Solution

1. Identify the integration method as Integration by Parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the variables for substitution by letting u=x and d(v)=cos(4*x)*d(x)

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

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

5. Substitute these components into the Integration by Parts formula.

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

6. Evaluate the remaining integral (∫_^)(sin(4*x)/4*d(x)) which results in −cos(4*x)/16

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

7. Simplify the signs to reach the final expression.

Final Answer

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

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