Consider the unction
`f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt`
Which is not true for `int_(0)^(x)f(t)dt` gt?

Options

A. positive for all `x in (0,1)`
B. increasing for all `x in (0,1)`
C. non-monotonic for all `x in (0,1)`
D. none of these

Answer

C

Explanation

`f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt.`
`rArr" "f'(x)=5ln(1+x^(2))-10x tan^(-1)x+16 sinx`
`rArr" "f''(x)=2(8 cos x-5 tan^(-1)x)`
`rArr" "f''(x)=-2(8sinx+(5)/(1+x^(2)))lt0AAx in (0,1)`
So, f''(x) is decreasing `AA x in (0,1)`
`rArr" "f''(x)gtf''(1)=2(8cos1-(5pi)/(4))`
`" "gt2(8cos.(pi)/(3)-(5pi)/(4))`
`" "=2(4-(5pi)/(4))gt0`
So, f''(x) is increasing, for `x gt 0 , f'(x)gtf'(0)=0`
So, f(x) is increasing, for `x gt0, f(x) gt f(0)=0`
So, `int_(0)^(x)f(t)` is positive and increasing.