JEE Advanced 2020 Full Test 5 Paper 1

‌ If ‌

b

∫

a

f(x)dx|=

b

∫

a

|f(x)∣dx
⇒f(x) is either positive in (a,b)
or f(x) is negative in (a,b) and also b>a
Also g′(x)=f(x)
When f(x)>0⇒g′(x)>0⇒g(x) is increasing function
When f(x)<0⇒g′(x)<0⇒g(x) is decreasing function
also g(0)=0

b

∫

a

f(x)g(x)dx=

b

∫

a

g(x)g′(x)dx
=‌

(g(b))2−(g(a))2

2

which will be positive for 0<a<b.