How to integrate x^2*sin(x)?

To integrate x^2*sin(x), use integration by parts.

Integration by parts is a technique used to integrate the product of two functions. It involves choosing one function to differentiate and the other to integrate. The formula for integration by parts is:

∫u dv = uv - ∫v du

where u and v are functions of x, and du/dx and dv/dx are their respective derivatives.

To integrate x^2*sin(x), let u = x^2 and dv/dx = sin(x). Then, du/dx = 2x and v = -cos(x) (by integrating sin(x)).

Using the formula for integration by parts, we have:

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

To integrate 2x*cos(x), let u = 2x and dv/dx = cos(x). Then, du/dx = 2 and v = sin(x) (by integrating cos(x)).

Using the formula for integration by parts again, we have:

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

Simplifying, we get:

∫x^2*sin(x) dx = -x^2*cos(x) + 2x*sin(x) + 2*cos(x) + C

where C is the constant of integration.