∫[f(x)g′(x)+g(x)f′(x)]f(x).g(x)[logf(x)+logg(x)]dx is equal to
A
f(x)g(x)log(f(x)g(x)+C
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B
12[logf(x)g(x)]2+C
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C
[logf(x)g(x)]2+C
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D
logf(x).g(x)+C
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Solution
The correct option is B12[logf(x)g(x)]2+C ∫[f(x)g′(x)+g(x)f′(x)]f(x).g(x)[logf(x)+logg(x)]dx=∫[f(x)g′(x)+g(x)f′(x)]f(x).g(x)[logf(x)g(x)]dx=∫logttdt[puttingf(x).g(x)=t⇒[f(x)g′x+g(x)f′(x)dx=dt]=12(logt)2+C=12[logf(x).g(x)]2+C