Calculus.mathlife |work| -

[ \int_a^b f(x) , dx = F(b) - F(a) ]

[ \fracddx \int_a^x f(t) , dt = f(x) ]

Differentiation and integration are inverse operations. calculus.mathlife

| Integral ( \int f(x) , dx ) | Result (plus constant ( C )) | | :--- | :--- | | ( \int x^n , dx ) (n ≠ -1) | ( \fracx^n+1n+1 ) | | ( \int \frac1x , dx ) | ( \ln |x| ) | | ( \int e^x , dx ) | ( e^x ) | | ( \int \cos x , dx ) | ( \sin x ) | | ( \int \sin x , dx ) | ( -\cos x ) | This theorem connects the two pillars. It says:

[ f'(x) = \lim_h \to 0 \fracf(x+h) - f(x)h ] [ \int_a^b f(x) , dx = F(b) -

Meaning: If you integrate a function and then differentiate the result, you get back the original function.

[ \int_a^b f(x) , dx = \lim_n \to \infty \sum_i=1^n f(x_i^*) \Delta x ] [ \int_a^b f(x) , dx = \lim_n \to

To compute a definite integral (total accumulation), evaluate the antiderivative at the endpoints and subtract.