What is Newton Leibnitz formula with Examples ?

Newton Leibnitz formula

If h(x) and g(x) are differentiable functions of x then,

\(d\over dx\) \(\int_{g(x)}^{h(x)}\) f(t)dt = f[h(x)].h'(x) – f[g(x)].g'(x)

Example : Evaluate \(d\over dt\) \(\int_{t^2}^{t^3}\) \(1\over log x\) dx

Solution : We have,

\(d\over dt\) \(\int_{t^2}^{t^3}\) \(1\over log x\) dx = \(1\over log t^3\) \(d\over dt\) \((t^3)\) – \(1\over log t^2\) \(d\over dt\) \((t^2)\)

= \(3t^2\over 3 log t\) – \(2t\over 2 log t\) = \(t(t – 1)\over log t\)

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