Evaluate the Integral

Problem

(∫_^)((x2+3*x)*cos(x)*d(x))

Solution

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

2. Choose u=x2+3*x and d(v)=cos(x)*d(x)

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

4. Apply the integration by parts formula for the first time:

(∫_^)((x2+3*x)*cos(x)*d(x))=(x2+3*x)*sin(x)−(∫_^)((2*x+3)*sin(x)*d(x))

5. Apply integration by parts again for the remaining integral (∫_^)((2*x+3)*sin(x)*d(x)) by setting (u_2)=2*x+3 and d((v_2))=sin(x)*d(x)

6. Differentiate (u_2) to find d((u_2))=2*d(x) and integrate d((v_2)) to find (v_2)=−cos(x)

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

(∫_^)((2*x+3)*sin(x)*d(x))=(2*x+3)*(−cos(x))−(∫_^)(−cos(x)*(2)*d(x))

8. Simplify the expression:

(∫_^)((2*x+3)*sin(x)*d(x))=−(2*x+3)*cos(x)+2*(∫_^)(cos(x)*d(x))

9. Integrate the final term:

(∫_^)((2*x+3)*sin(x)*d(x))=−(2*x+3)*cos(x)+2*sin(x)

10. Combine all parts back into the original equation, distributing the negative sign:

(∫_^)((x2+3*x)*cos(x)*d(x))=(x2+3*x)*sin(x)−(−(2*x+3)*cos(x)+2*sin(x))+C

11. Simplify the final expression by grouping terms:

(∫_^)((x2+3*x)*cos(x)*d(x))=(x2+3*x−2)*sin(x)+(2*x+3)*cos(x)+C

Final Answer

(∫_^)((x2+3*x)*cos(x)*d(x))=(x2+3*x−2)*sin(x)+(2*x+3)*cos(x)+C

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