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Pattern – 2 : Definite integration

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⋆ Pattern – 2

The value of a definite integral does not change with change of variable of integration provided the limits of integration remains the same.

i.e. $\int_{a}^{b}f\left(x\right)\ dx$ = $\int_{a}^{b}f\left(t\right)\ dt$ = $\int_{a}^{b}f\left(u\right)\ du$

Proof :

Let, ∫ f(x) dx = F (x) + c

$\int_{a}^{b}f\left(x\right)dx=\left[F\left(x\right)\right]_a^b$

= F(b) – F(a)

$\int_{a}^{b}{f\left(t\right)\ dt}=\ \left[F\left(t\right)\right]_a^b$

= F(b) – F(a)

$\int_{a}^{b}f\left(u\right)\ du=\ \left[F\left(u\right)\right]_a^b$

= F(b) – F(a)

$x^2+1\rightarrow t$

$\int_{a}^{b}\left(2t+7\right)\ dt\rightarrow\ \left[t^2+7t\right]_a^b$

$=\ \int_{a}^{b}\left(2x+7\right)\ dx\rightarrow\ \left[x^2+7x\right]_a^b$