Step 1: Recognize the Integral

We want to find the integral of the expression x sin(x²): ∫ x sin(x²) dx

Step 2: Apply a u-Substitution

Let u = x². Then, du = 2x dx. We can rewrite the integral as:

∫ (1/2) sin(u) du

Step 3: Integrate the Expression in Terms of u

The integral of sin(u) is -cos(u), so we have:

∫ (1/2) sin(u) du = -(1/2) cos(u) + C

Step 4: Substitute Back for x

Substitute back u = x²:

-(1/2) cos(x²) + C

Conclusion: The integral of x sin(x²) is -(1/2) cos(x²) + C. This result is obtained using a u-substitution, a powerful technique in integral calculus.