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

Problem

(∫_^)(x2*sin(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)=sin(x)*d(x)

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

4. Substitute these into the integration by parts formula.

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

5. Simplify the expression by pulling out the constant.

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

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

7. Differentiate u to find d(u)=d(x) and integrate d(v) to find v=sin(x)

8. Substitute these into the formula for the inner integral.

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

9. Evaluate the final integral (∫_^)(sin(x)*d(x))=−cos(x)

(∫_^)(x*cos(x)*d(x))=x*sin(x)+cos(x)

10. Combine all parts back into the main equation and add the constant of integration C

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

11. Distribute the constant to reach the final simplified form.

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

Final Answer

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

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