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  4. Let F (x)=f (x)+f((1/x)), where f(x)=∫ limitsx1(log t/1+t)dt. Then F(e) equals

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Q. Let $F \left(x\right)=f \left(x\right)+f\left(\frac{1}{x}\right),$ where $f(x)=\int\limits^{x}_{{1}}$$\frac{log\,t}{1+t}dt$. Then $F(e)$ equals

AIEEEAIEEE 2007Integrals

Solution:

$f(x)=\int\limits^{x}_{{1}}$$\frac{log\,t}{1+t}dt$
$f \left(e\right)=f \left(e\right)+f \left(\frac{1}{e}\right)$
$f \left(e\right)=\int\limits^{e}_{{1}}$$\frac{log\,t}{1+t}dt$$+\int\limits^{1/e}_{{1}}$$\frac{log\,t}{1+t}dt$
$=\int\limits^{e}_{{1}}\frac{log\,t}{1+t}dt+\int\limits^{e}_{{1}}$$\frac{log\,t}{t\left(1+t\right)}dt$
$=\int\limits^{e}_{{1}}$$\frac{log\,t}{1+t}dt$$=\frac{1}{2}.$