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Question

[f(x)g(x)f(x)g(x)f(x)g(x)].[log(g(x))log(f(x))]dx=

A
logg(x)f(x)+c
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B
12(logg(x)f(x))2+c
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C
g(x)f(x)log(g(x)f(x))+c
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D
f(x)g(x)log(g(x)f(x))+c
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Solution

The correct option is B 12(logg(x)f(x))2+c
[f(x)g(x)f(x)g(x)f(x)g(x)][logg(x)logf(x)]dx=I
I=f(x)g(x)f(x)g(x)f2(x)f(x)g(x)logg(x)f(x)dx
I=[f(x)g(x)f(x)g(x)f2(x)]logg(x)f(x)g(x)/f(x)dx
Let logg(x)f(x)=u
1g(x)f(x)[f(x)g(x)f(x)g(x)f2(x)]dx=du
I=udu
I=u22+c
I=12(logg(x)f(x))2+c.

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