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Pattern – 3 : Definite integration

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⋆ Pattern – 3

$\int_{a}^{b}f\left(x\right)\ dx=\ -\ \int_{b}^{a}f\left(x\right)\ dx$

Proof :

Let, ∫ f(x) dx = F(x) + c

LHS = $\int_{a}^{b}f\left(x\right)dx$

$= \left[F\left(x\right)\right]_a^b$

= F(b) – F(a) —(1)

RHS = – $\int_{b}^{a}f \left(x\right)dx$

= – $\left[F\left(x\right)\right]_b^a$

= – [ F(a) – F (b) ]

= F(b) – F(a) —–(2)

∴ LHS = RHS

$\therefore\ \ \int_{a}^{b}f\left(x\right)dx=\ -\int_{b}^{a}f\left(x\right)\ dx$