What is the result of ∫ x sin^2(x) dx using integration by parts?

The integral of ( \int x \sin^2(x) , dx ) can be solved using integration by parts. Integration by parts is based on the formula:

[

\int u , dv = uv - \int v , du

]

Here, we need to choose ( u ) and ( dv ) wisely. A common choice for ( u ) when dealing with products involving polynomials and trigonometric functions is to set:

• ( u = x ), which gives ( du = dx )

• ( dv = \sin^2(x) , dx )

To use this successfully, we first need to integrate ( dv ). The integral ( \int \sin^2(x) , dx ) can be rewritten using the identity:

[

\sin^2(x) = \frac{1 - \cos(2x)}{2}

]

This means that:

[

\int \sin^2(x) , dx = \frac{1}{2} \int (1 - \cos(2x)) , dx = \frac{1}{2} \left( x - \frac{1}{2} \sin(