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

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