How to Integrate x^2 cos x?

To integrate the function x² cos x, we can use the method of integration by parts, which is based on the formula:

∫u dv = uv – ∫v du

Here, we will choose:

• u = x² (which means du = 2x dx)
• dv = cos x dx (which means v = sin x)

Now applying the integration by parts formula:

∫x² cos x dx = x² sin x – ∫sin x (2x) dx

The remaining integral, ∫sin x (2x) dx, still requires integration by parts. So, we apply it again:

• u = 2x (thus du = 2 dx)
• dv = sin x dx (therefore v = -cos x)

Using the integration by parts formula again:

∫(2x) sin x dx = -2x cos x + ∫2 cos x dx

The integral of cos x is straightforward:

∫2 cos x dx = 2 sin x

Now putting everything together:

∫x² cos x dx = x² sin x – (-2x cos x + 2 sin x)

So, we simplify that:

∫x² cos x dx = x² sin x + 2x cos x – 2 sin x + C

Here, C is the constant of integration. Therefore, the final answer is:

∫x² cos x dx = x² sin x + 2x cos x – 2 sin x + C