[ \int_a^b f(x) , dx = \lim_n \to \infty \sum_i=1^n f(x_i^*) \Delta x ]
Where ( F ) is any antiderivative of ( f ).
| Function ( f(x) ) | Derivative ( f'(x) ) | | :--- | :--- | | Constant ( c ) | 0 | | ( x^n ) | ( n x^n-1 ) | | ( e^x ) | ( e^x ) | | ( \ln x ) | ( 1/x ) | | ( \sin x ) | ( \cos x ) | | ( \cos x ) | ( -\sin x ) | Core Question: What total amount builds up from a continuously changing rate? calculus.mathlife
Interpretation: We take two points on a curve, bring them infinitely close together, and measure the slope of the resulting tangent line.
| 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: [ \int_a^b f(x) , dx = \lim_n \to
Interpretation: We slice the area under a curve into infinitely thin rectangles, sum them up, and get the exact total.
Meaning: If you integrate a function and then differentiate the result, you get back the original function. | Integral ( \int f(x) , dx )
[ \int_a^b f(x) , dx = F(b) - F(a) ]