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Pattern – 24 : Definite integration

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⋆ Pattern – 24(Leibnitz’s Theorem)

If f is a continuous function on [a, b] and u(x) and v(x) are differentiable functions of x whose values lie in [a, b], then

\frac{d}{dx}\left\{\int_{u(x)}^{v(x)}f\left(t\right)\ \ dt\right\}

= f{v(x)} \frac{d}{dx} v(x) - f{u(x)} \frac{d}{dx} u(x)

Proof :

Let, \frac{d}{dx}F\left(x\right)=f\left(x\right) ——-(1)

∴  F(x) + c = ∫f(x) dx —-(2)

Now,

∫ f(t) dt = F(t) + c

\therefore\ \ \int_{u\left(x\right)}^{v\left(x\right)}f\left(t\right)dt=\ \left[F\left(t\right)\right]_{u\left(x\right)}^{v\left(x\right)}

= F(v(x)) – F(u(x))

\therefore\ \ \frac{d}{dx}\ \int_{u\left(x\right)}^{v\left(x\right)}f\left(t\right)dt= F^\prime\left(v\left(x\right)\right)\ \frac{d}{dx}v\left(x\right) – F^\prime\left(u\left(x\right)\right)\frac{du}{dx}\left(x\right)

= f\ \left(v\left(x\right)\right)\frac{d}{dx}v\left(x\right) – f\left(u\left(x\right)\right)\frac{d}{dx}u\left(x\right)

 

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