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

Problem

(∫_^)(x*sin(2*x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 2*x2 is a multiple of the outer factor x

2. Define the substitution variable u=2*x2

3. Differentiate u with respect to x to find d(u)=4*x*d(x)

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

5. Substitute the variables into the integral to rewrite it in terms of u

(∫_^)(sin(u)⋅1/4*d(u))

6. Factor out the constant from the integral.

1/4*(∫_^)(sin(u)*d(u))

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

−1/4*cos(u)+C

8. Back-substitute the original expression for u to get the final result in terms of x

−1/4*cos(2*x2)+C

Final Answer

(∫_^)(x*sin(2*x2)*d(x))=−1/4*cos(2*x2)+C

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