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

Problem

(∫_^)(x*sin(x2)*d(x))

Solution

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

2. Substitute u=x2 to simplify the integrand.

3. Differentiate u with respect to x to find d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

4. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(x*sin(x2)*d(x))=(∫_^)(sin(u)1/2*d(u))

5. Factor out the constant 1/2 from the integral.

1/2*(∫_^)(sin(u)*d(u))

6. Integrate the function sin(u) with respect to u recalling that the integral of sin(u) is −cos(u)

−1/2*cos(u)+C

7. Back-substitute the original expression x2 for u to obtain the final result in terms of x

Final Answer

(∫_^)(x*sin(x2)*d(x))=−1/2*cos(x2)+C

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