Evaluate the Integral integral of x^2*cos(x) with respect to x

Problem

(∫_^)(x2*cos(x)*d(x))

Solution

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

2. Choose the variables for the first application of integration by parts by letting u=x2 and d(v)=cos(x)*d(x)

3. Differentiate and integrate to find d(u)=2*x*d(x) and v=sin(x)

4. Apply the integration by parts formula:

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)−(∫_^)(2*x*sin(x)*d(x))

5. Apply integration by parts a second time for the remaining integral (∫_^)(2*x*sin(x)*d(x)) by letting u=2*x and d(v)=sin(x)*d(x)

6. Differentiate and integrate to find d(u)=2*d(x) and v=−cos(x)

7. Substitute these values into the second integration by parts step:

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

8. Evaluate the simple integral (∫_^)(2*cos(x)*d(x))=2*sin(x)

9. Combine all parts and simplify the signs:

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)−(−2*x*cos(x)+2*sin(x))+C

10. Distribute the negative sign to reach the final expression.

Final Answer

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)+2*x*cos(x)−2*sin(x)+C

Want more problems? Check here!