Fundamental

JEE Mains 2015 Sample Questions › Fundamentals questions › Fundamental

$Let \, \, y= f(x)\, \, \textrm{be continous ,positive,and increasing over the interval [a,b] }$

$Show \,\, that\, \,\rightarrow \boxed{\, \, \int_{a}^{b}f(x)dx+\int_{f(a)}^{f(b)}f^{-1}(y)dy=bf(b)-af(a)}$

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Let $t = f (z)$ Then $\int x y f - 1 (t) d t = \int f - 1 (x) f - 1 (y) z f , (z) d z$ $= (z f (z)) ∣ f - 1 (x) f - 1 (y) - \int f - 1 (x) f - 1 (y) f (z) d z$ (integrating by parts)

So, we get $\int x y f - 1 (t) d t = y f - 1 (y) - x f - 1 (x) - \int f - 1 (x) f - 1 (y) f (t) d t$

Substituting $x = f (a), y = f (b)$ we get $\boxed{\, \, \int_{a}^{b}f(x)dx+\int_{f(a)}^{f(b)}f^{-1}(y)dy=bf(b)-af(a)}$

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Shubhodip ·Jul 6 '11 at 6:35

just found from nishant sir's bookmark..

http://www.targetiit.com/iit-jee-forum/posts/e-x-2-1849.html

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Basic Integral Calculus Operators Symbols Matrix Cases

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° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω

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Fundamental

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