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Pattern – 4 : Definite integration

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$\int_{a}^{b}f\left[g\left(x\right)\right]g^\prime\left(x\right)dx=\ \int_{g\left(a\right)}^{g\left(b\right)}f\left(t\right)dt$

Proof :

Let, g(x) = t

∴ g’ (x) dx = dt

x = a ⇒ t = g(a)

x = b ⇒ t = g(b)

$\therefore\ \ \int_{a}^{b}f\left[g\left(x\right)\right]\ g^\prime\left(x\right)\ dx=\ \int_{g\left(a\right)}^{g\left(b\right)}g\left(t\right)\ dt$