# Doubt from Definite Integration

**Viram_2019**#2

Observe that

(1+x^2)^2 = (1+x^4)+2x^2

For all x \geq 0 we can say

(1+x^2)^2 \geq 1+x^4

\cfrac{1}{(1+x^2)^2} \leq \cfrac{1}{1+x^4}

\Rightarrow \cfrac{1}{1+x^2} \leq \cfrac{1}{\sqrt{1+x^4}}

Integrating preserves the inequality hence integrate both sides with limits from 0 to 1

\displaystyle \int_\limits{0}^1 \cfrac{1}{1+x^2} \leq \int_\limits{0}^1 \cfrac{1}{\sqrt{1+x^4}}

\Rightarrow I \geq \tan^{-1} (1)

\Rightarrow I \geq \cfrac{\pi}{4}

One can easily see that for the first condition integral of sinx, a and b will have a difference of 4π. And from the second condition we can say that 0 and (a+b) will have a difference of 4.5π.

So

b-a=4π

a+b = 4.5π

**Sneha_2021**#11

apply lebniz but first apply definate integral property

fx=f(a+b-x) then apply product rule of differentiation

. @Gaurav_2020_3

**Tanmay_1**#20

I was in confusion with the interval of x given in option in answer

for increasing and decreasing function and the interval obtained

By me...that mens i am correct in finding interval for increasing

And decresing pls reply