Final answer:
The integral of 7xcos(x²) is (7/2) sin(x²) + C, and the correctness of the result is confirmed by differentiating it to retrieve the original integrand. We used a substitution method to find the solution.
Step-by-step explanation:
To find the integral of 7xcos(x²), we can use the substitution method. Let u = x², which implies that du = 2xdx. Therefore, dx = du/2x. We can substitute into the integral to get:
- ∫ 7x cos(x²) dx = ∫ 7x cos(u) × (du/2x) = ∫ (7/2) cos(u) du
- The integral of cos(u) is sin(u), so we get (7/2) sin(u) + C.
- Substituting back for u, we have (7/2) sin(x²) + C.
To check our answer through differentiation, we differentiate (7/2) sin(x²) with respect to x to get:
(7/2) × 2x cos(x²) = 7x cos(x²), which is our original integrand. Therefore, the integration is correct.