DEFINITE INTEGRALS

If f:Râ†’R is a monotonic,differentiable real valued function a,b are two real numbers and

âˆ«f(x)+f(a))(f(x)-f(a))dx=I₁
upper limitâ†’ b and lower limitâ†’a

âˆ«x(b-f^-1(x))dx=I₂

upper limitâ†’f(b)
lower limitâ†’f(a)

I₂/I₁=____

2 Answers

neone there

Applying uv rule in first integral

I1=

[f(b)²-f(a)²]b - _aâˆ«^b2xf(x)f'(x)dx

for I2

as f(x) is monotonic we can take x=f(k) and put the limits a and b and further

f^-1(f(k)) then = k

so we get it as
_aâˆ«^bf(k)f'(k)(b-k)dk

we can safely put k as x

so
_aâˆ«^bf(x)f'(x)(b-x)dx

b_aâˆ«^bf(x)f'(x)dx-_aâˆ«^bxf(x)f'(x)dx

putting f(x)=t in the first expression we get finally

b[f(b)²-f(a)²]/2 - _aâˆ«^bxf(x)f'(x)dx

thus clearly I1/I2=2

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