## Example 7.2.6(b): Applying Integration by Parts

This time we defineThen

g'(x) = cos(x)so thatg(x) = cos(x) dx = sin(x)f(x) = xso that^{2}f'(x) = 2x

*G(x) = f(x) g(x) = sin(x) x*and

^{2}Intergration by parts reduced the original problem to finding a slightly simpler integral, but we can not find the second integral immediately. What worked once might work again so let's use integration by parts on the second integral with:x^{2}cos(x) dx = G(b) - G(a) - 2x sin(x) dx

so that

g'(x) = sin(x)so thatg(x) = sin(x) dx = -cos(x)f(x) = 2xso thatf'(x) = 2

*G*. Therefore

^{*}(x) = -2x cos(x)Taking everything together we have:2x sin(x) dx = G^{*}(b) - G^{*}(a) + 2 cos(x) dx

x^{2}cos(x) dx = G(b) - G(a) - [G^{*}(b) - G^{*}(a) + 2 cos(x) dx ] =

= G(b) - G(a) - G^{*}(b) + G^{*}(a) - 2 (sin(b) - sin(a)) =

= sin(b) b^{2}- sin(a) a^{2}+ 2b cos(b) - 2a cos(a) -2sin(b) + 2sin(a)