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

Question

Let F (x)=f (x)+f((1/x)), where f(x)=∫ limitsx1(log t/1+t)dt. Then F(e) equals (A) (1/2) (B) 0 (C) 1 (D) 2

Tardigrade Admin · Accepted Answer

f(x)=∫ limitsx1(log t/1+t)dt f (e)=f (e)+f ((1/e)) f (e)=∫ limitse1(log t/1+t)dt+∫ limits1/e1(log t/1+t)dt =∫ limitse1(log t/1+t)dt+∫ limitse1(log t/t(1+t))dt =∫ limitse1(log t/1+t)dt=(1/2).

Q. Let $F (x) = f (x) + f (\frac{1}{x}),$ where $f (x) = 1 \int x$ $\frac{l o g t}{1 + t} d t$ . Then $F (e)$ equals

$\frac{1}{2}$

$0$

$1$

$2$

Q. Let F(x)=f(x)+f(x1​), where f(x)=1∫x​1+tlogt​dt. Then F(e) equals

21​

0

1

2

f(x)=1∫x​1+tlogt​dt f(e)=f(e)+f(e1​) f(e)=1∫e​1+tlogt​dt+1∫1/e​1+tlogt​dt =1∫e​1+tlogt​dt+1∫e​t(1+t)logt​dt =1∫e​1+tlogt​dt=21​.

Q. Let $F (x) = f (x) + f (\frac{1}{x}),$ where $f (x) = 1 \int x$ $\frac{l o g t}{1 + t} d t$ . Then $F (e)$ equals

$\frac{1}{2}$

$0$

$1$

$2$