Evaluate the Integral integral of x^2sin(3x) with respect to x

Problem

(∫_^)(x2*sin(3*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 and d(v)=sin(3*x)*d(x)

3. Calculate the differentials d(u)=2*x*d(x) and v=−1/3*cos(3*x)

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

(∫_^)(x2*sin(3*x)*d(x))=−1/3*x2*cos(3*x)−(∫_^)(−2/3*x*cos(3*x)*d(x))

5. Simplify the expression before the second integration.

(∫_^)(x2*sin(3*x)*d(x))=−1/3*x2*cos(3*x)+2/3*(∫_^)(x*cos(3*x)*d(x))

6. Apply integration by parts again for the remaining integral (∫_^)(x*cos(3*x)*d(x)) with u=x and d(v)=cos(3*x)*d(x)

7. Calculate the new differentials d(u)=d(x) and v=1/3*sin(3*x)

(∫_^)(x*cos(3*x)*d(x))=1/3*x*sin(3*x)−(∫_^)(1/3*sin(3*x)*d(x))

8. Evaluate the final integral (∫_^)(1/3*sin(3*x)*d(x))=−1/9*cos(3*x)

(∫_^)(x*cos(3*x)*d(x))=1/3*x*sin(3*x)+1/9*cos(3*x)

9. Substitute this result back into the main equation.

(∫_^)(x2*sin(3*x)*d(x))=−1/3*x2*cos(3*x)+2/3*(1/3*x*sin(3*x)+1/9*cos(3*x))+C

10. Distribute and combine terms to find the final result.

(∫_^)(x2*sin(3*x)*d(x))=−1/3*x2*cos(3*x)+2/9*x*sin(3*x)+2/27*cos(3*x)+C

Final Answer

(∫_^)(x2*sin(3*x)*d(x))=−1/3*x2*cos(3*x)+2/9*x*sin(3*x)+2/27*cos(3*x)+C

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