Question

Can somebody tell me how to find the anti-derivative of
cos(x) * x

and

cos(x) * sin(x)

Answer 1

to integrate cos(x) * x, I'd use the Integration by Parts rule

http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx

Answer 2

to integrate cos(x) * sin(x), use the identity

sin(2x) = 2*sin(x)*cos(x)

Answer 3

then use u-substitution

Answer 4

The first one requires a clever use of the product rule. The second one requires a clever use of the chain rule.

product rule
(f*g)'=f'g+fg'

chain rule
(f(g))'=f'(g)*g'