Evaluate the Integral

Problem

(∫_^)((x2+9*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+9*x and d(v)=cos(x)*d(x)

3. Differentiate u to find d(u)=(2*x+9)*d(x) and integrate d(v) to find v=sin(x)

4. Apply the formula for the first time:

(∫_^)((x2+9*x)*cos(x)*d(x))=(x2+9*x)*sin(x)−(∫_^)((2*x+9)*sin(x)*d(x))

5. Apply integration by parts again for the remaining integral (∫_^)((2*x+9)*sin(x)*d(x)) by choosing (u_2)=2*x+9 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 into the second integration by parts step:

(∫_^)((2*x+9)*sin(x)*d(x))=−(2*x+9)*cos(x)−(∫_^)(−2*cos(x)*d(x))

8. Simplify the second integral:

(∫_^)((2*x+9)*sin(x)*d(x))=−(2*x+9)*cos(x)+2*sin(x)

9. Combine all parts back into the original equation:

(∫_^)((x2+9*x)*cos(x)*d(x))=(x2+9*x)*sin(x)−(−(2*x+9)*cos(x)+2*sin(x))+C

10. Distribute the negative sign and simplify the expression:

(∫_^)((x2+9*x)*cos(x)*d(x))=(x2+9*x)*sin(x)+(2*x+9)*cos(x)−2*sin(x)+C

11. Factor the sin(x) terms:

(∫_^)((x2+9*x)*cos(x)*d(x))=(x2+9*x−2)*sin(x)+(2*x+9)*cos(x)+C

Final Answer

(∫_^)((x2+9*x)*cos(x)*d(x))=(x2+9*x−2)*sin(x)+(2*x+9)*cos(x)+C

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