The antiderivative of 2x cos(x^2) dx is which expression?

• A. cos(x^2) + C
• B. cos(2x) + C
• C. sin(x^2) + C
• D. -sin(x^2) + C

Answer: (C)

This uses the idea of reversing the chain rule with substitution. If you set u = x^2, then du = 2x dx, so the integral ∫ 2x cos(x^2) dx becomes ∫ cos(u) du, which integrates to sin(u) + C. Substituting back gives sin(x^2) + C. The derivative of sin(x^2) is cos(x^2) times the inner derivative 2x, which matches the integrand exactly. Other options don’t work because differentiating them yields different expressions (for example, d/dx[cos(x^2)] = -2x sin(x^2), d/dx[cos(2x)] = -2 sin(2x), and d/dx[-sin(x^2)] = -2x cos(x^2)). The correct antiderivative is sin(x^2) + C.