Question

Find the antiderivative of xcos^2(x^2) dx.

Answer 1

To find the antiderivative of x cos²(x²) dx, we can use a substitution method. Substituting u = x², we can evaluate the integral and simplify to get the final answer: (1/4) x² + (sin(2x²)/4) + C.

Step-by-step explanation:

To find the antiderivative of x cos²(x²) dx, we can use a substitution. Let's substitute u = x². Then, du = 2x dx. We can rearrange this to get dx = du / (2x).

Substituting these into the original integral, we get:
∫ x cos²(x²) dx = ∫ (1/2) cos²(u) du

Now, we know that the antiderivative of cos²(u) is (u/2) + (sin(2u)/4). So, substituting back we get:
∫ (1/2) cos²(u) du = (1/2) * (u/2) + (sin(2u)/4) + C.
Finally, substitute back u = x² to get the final answer: (1/4) x² + (sin(2x²)/4) + C.