Integrate y=(x^2)cos(x) with respect to x.

This problem must be solved using integration by parts, since y is equal to the product of two functions of x. Let u=x^2, therefore u'=2x (u'=du/dx). Let v'=cos(x), therefore v=sin(x) (v'=dv/dx).Using the integration by parts formula: Intgrl(y)dx= uv - Intgrl(vu')dx (which is given on the A Level maths formula sheet), Intgrl(y)dx=x^2(sin(x)) - Intgrl(2x(sin(x))dx). We must apply the same rule again, since 2x(sin(x)) cannot be integrated directly. u=2x, u'=2v'=sin(x), v=-cos(x). Intgrl(y)dx=x^2(sin(x)) - [-2x(cos(x)) - intgrl(2(-cos(x))dx], -2cos(x) can now be integrated giving the final solution: Intgrl(y)=x^2(sin(x)) + 2x(cos(x)) - 2(sin(x)) + c.