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

Problem

(∫_^)(x*sin(2*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 components for the formula by letting u=x and d(v)=sin(2*x)*d(x)

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

4. Integrate d(v) to find v=−1/2*cos(2*x)

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

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

6. Simplify the expression and the integral.

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

7. Evaluate the remaining integral.

(∫_^)(cos(2*x)*d(x))=1/2*sin(2*x)

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

(∫_^)(x*sin(2*x)*d(x))=−x/2*cos(2*x)+1/4*sin(2*x)+C

Final Answer

(∫_^)(x*sin(2*x)*d(x))=−x/2*cos(2*x)+1/4*sin(2*x)+C

Want more problems? Check here!