Evaluate the Integral integral of xsin(3x) with respect to x

Problem

(∫_^)(x*sin(3*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(3*x)*d(x)

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

4. Integrate d(v) to find v=−1/3*cos(3*x)

5. Apply the formula by substituting u v d(u) and d(v) into the integration by parts equation.

(∫_^)(x*sin(3*x)*d(x))=−x/3*cos(3*x)−(∫_^)(−1/3*cos(3*x)*d(x))

6. Simplify the integral term by pulling out the constant.

(∫_^)(x*sin(3*x)*d(x))=−x/3*cos(3*x)+1/3*(∫_^)(cos(3*x)*d(x))

7. Evaluate the remaining integral (∫_^)(cos(3*x)*d(x))=1/3*sin(3*x)

(∫_^)(x*sin(3*x)*d(x))=−x/3*cos(3*x)+1/3*(1/3*sin(3*x))+C

8. Simplify the final expression and add the constant of integration C

Final Answer

(∫_^)(x*sin(3*x)*d(x))=−x/3*cos(3*x)+1/9*sin(3*x)+C

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