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

Problem

(∫_^)(x*sin(3*x2)*d(x))

Solution

1. Identify the substitution method as the best approach because the derivative of the inner function 3*x2 is a multiple of the outer factor x

2. Substitute u=3*x2 to simplify the integrand.

3. Differentiate u with respect to x to find d(u)=6*x*d(x)

4. Rearrange the differential to solve for the terms present in the integral: 1/6*d(u)=x*d(x)

5. Rewrite the integral in terms of u

(∫_^)(sin(u)⋅1/6*d(u))

6. Factor out the constant:

1/6*(∫_^)(sin(u)*d(u))

7. Integrate the sine function using the rule (∫_^)(sin(u)*d(u))=−cos(u)+C

−1/6*cos(u)+C

8. Back-substitute u=3*x2 to return to the original variable x

Final Answer

(∫_^)(x*sin(3*x2)*d(x))=−1/6*cos(3*x2)+C

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