Evaluate the Integral

Problem

(∫_^)((x2+5*x)*cos(x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x2+5*x and d(v)=cos(x)*d(x)

2. Differentiate u to find d(u)=(2*x+5)*d(x) and integrate d(v) to find v=sin(x)

3. Apply the integration by parts formula for the first time.

(∫_^)((x2+5*x)*cos(x)*d(x))=(x2+5*x)*sin(x)−(∫_^)((2*x+5)*sin(x)*d(x))

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

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

6. Substitute these into the second integration by parts step.

(∫_^)((2*x+5)*sin(x)*d(x))=−(2*x+5)*cos(x)−(∫_^)(−2*cos(x)*d(x))

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

(∫_^)((2*x+5)*sin(x)*d(x))=−(2*x+5)*cos(x)+2*sin(x)

8. Combine all parts back into the original expression, ensuring to distribute the negative sign.

(∫_^)((x2+5*x)*cos(x)*d(x))=(x2+5*x)*sin(x)−(−(2*x+5)*cos(x)+2*sin(x))+C

9. Simplify the expression by grouping terms.

(∫_^)((x2+5*x)*cos(x)*d(x))=(x2+5*x−2)*sin(x)+(2*x+5)*cos(x)+C

Final Answer

(∫_^)((x2+5*x)*cos(x)*d(x))=(x2+5*x−2)*sin(x)+(2*x+5)*cos(x)+C

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