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Let f(x)= 0x g(t) e ( 1-t1+t )dt, where g is a continuous odd function. If - /2 /2 (f(x)+ x2 x1+ex )dx= ( )2- , then is equal to .
2
g(t) odd, e 1-t1+t odd integrand even f(x)= 0x(even) dt is odd
- /2 /2f(x) dx=0
I= - /2 /2 x2 x1+exdx; x -x: I= - /2 /2 ex x2 x1+exdx
Adding: 2I= - /2 /2x2 x dx I= 0 /2x2 x dx
= [x2 x+2x x-2 x ]0 /2= 24-2 (by parts twice)
= ( 2 )2-2 =2
Answer: 2
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