What is f(x) = int 2sinx-xcosx dxf(x)=∫2sinx−xcosxdx if f((7pi)/6) = 0 f(7π6)=0?

1 Answer
Teddy
Jun 14, 2018

f(x)=-3cosx-xsinx-(3sqrt3)/2-(7pi)/12f(x)=−3cosx−xsinx−3√32−7π12

Explanation:

Split up the integral into
f(x)=2intsinxdx-intxcosxdxf(x)=2∫sinxdx−∫xcosxdx
Using integration by parts, we get
f(x)=-2cosx-xsinx-cosx+Cf(x)=−2cosx−xsinx−cosx+C
f(x)=-3cosx-xsinx+Cf(x)=−3cosx−xsinx+C
Substituting x=(7pi)/6x=7π6, we get
f((7pi)/6)=0=-3cos((7pi)/6)-(7pi)/6sin((7pi)/6)+Cf(7π6)=0=−3cos(7π6)−7π6sin(7π6)+C
Solving for C, we get
C=-(3sqrt3)/2-(7pi)/12C=−3√32−7π12
Therefore, f(x)=-3cosx-xsinx-(3sqrt3)/2-(7pi)/12f(x)=−3cosx−xsinx−3√32−7π12.

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