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

Problem

(∫_^)(x*cos(2*x)*d(x))

Solution

1. Identify the method of integration by parts, which is defined by the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the components for the formula by letting u=x and d(v)=cos(2*x)*d(x)

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

4. Integrate d(v) to find v=(∫_^)(cos(2*x)*d(x))=1/2*sin(2*x)

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

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

6. Evaluate the remaining integral (∫_^)(1/2*sin(2*x)*d(x))

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

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

1/2*x*sin(2*x)−(−1/4*cos(2*x))+C

8. Simplify the signs.

1/2*x*sin(2*x)+1/4*cos(2*x)+C

Final Answer

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

Want more problems? Check here!