Intregration strategy

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Homework Statement


I(secx*cos2x/(sinx+secx),x)
I(cos2x/(sinxcosx+1),x)
I(cos2x/(sin2x/2+1),x)
u=sin2x du/2=cos2xdx
1/2 I( 1/ (u/2+1) , x)
I( 1/ (u+2) , x)
ln|u+2|+C=ln|sin2x+2|+C

also

I((x^2-bx)sin2x,x)
I(x^2sin2x,x)-bI(xsin2x,x)
I(x^2sin2x,x):
u=x^2 du= 2xdx
dv=sin2x v=-(1/2)cos2x
-x^2cos2x/2+I(xcos2x,x)
u=x du=dx
dv=cos2x v=sin2x/2
-x^2cos2x/2+xsin2x/2-I(sin2x,x)/2
-x^2cos2x/2+xsin2x/2+cos2x/4

-bI(xsin2x,x):
u=x du=dx
dv=sin2x v=-cos2x/2
-b[-xcos2x/2+I(cos2x,x)/2]
bxcos2x/2-bsin2x/4

thus:
I((x^2-bx)sin2x,x)=-x^2cos2x/2+xsin2x/2+cos2x/4+bxcos2x/2-bsin2x/4+C



Homework Equations





The Attempt at a Solution

 

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