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Question

State whether the given equation is true or false
(px+qx)2px.qx+e5logexe4logexe3logexe2logex+sin2x+cos2xsin2xcos2xdx
=(p/q)xlog(p/q)+(q/p)xlog(q/p)+2x+x33+tanxcotx.

A
True
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B
False
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Solution

The correct option is A True
Let I=(px+qx)2px.qx+e5logexe4logexe3logexe2logex+sin2x+cos2xsin2xcos2xdx
I=I1+I2+I3
Where
I1=(px+qx)2px.qxdx=(p2xpxqx+q2xpxqx+2pxqxpxqx)dx
=(1qx/px+1px/qx+2)dx=(p/q)xlog(p/q)+(q/p)xlog(q/p)+2x
I2=e5logexe4logexe3logexe2logexdx=x5x4x3x2dx=x2dx=x33
I3=sin2x+cos2xsin2xcos2xdx=(sec2x+csc2x)dx=tanxcotx
Hence
I=(p/q)xlog(p/q)+(q/p)xlog(q/p)+2x+x33+tanxcotx

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