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

Problem

(∫_^)(x*sin(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. Choose the variables for substitution by letting u=x and d(v)=sin(4*x)*d(x)

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

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

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

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

6. Simplify the expression and the integral:

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

7. Evaluate the remaining integral:

(∫_^)(cos(4*x)*d(x))=1/4*sin(4*x)

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

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

Final Answer

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

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