What is #f(x) = int 2sinx-xcosx dx# if #f((7pi)/6) = 0 #?

1 Answer
Teddy
Jun 14, 2018

#f(x)=-3cosx-xsinx-(3sqrt3)/2-(7pi)/12#

Explanation:

Split up the integral into
#f(x)=2intsinxdx-intxcosxdx#
Using integration by parts, we get
#f(x)=-2cosx-xsinx-cosx+C#
#f(x)=-3cosx-xsinx+C#
Substituting #x=(7pi)/6#, we get
#f((7pi)/6)=0=-3cos((7pi)/6)-(7pi)/6sin((7pi)/6)+C#
Solving for C, we get
#C=-(3sqrt3)/2-(7pi)/12#
Therefore, #f(x)=-3cosx-xsinx-(3sqrt3)/2-(7pi)/12#.

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