1.

Consider the unction f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt f(x) is

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negative for all `X in (0,1)`
increasing for all `x in (0,1)`
DECREASING for all `x in (0,1)`
non-monotonic function for `x in (0,1)`

Solution :`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.


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