Math2.org Math Tables: Integral Identities
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Formal Integral Definition:
![\[ \int_a^b f(x) \, dx = \lim<sub>d \to 0</sub> \sum<sub>k=1</sub><sup>n</sup> f(X_k)(x_k - x<sub>k-1</sub>) \]](/math/tex/91fcca28868e20477424a313a4f74c9a.png "\[ \int_a^b f(x) \, dx = \lim<sub>d \to 0</sub> \sum<sub>k=1</sub><sup>n</sup> f(X_k)(x_k - x<sub>k-1</sub>) \]")
when...
a = x0 < x1 < x2 < ... < xn = b
d = max (x1-x0, x2-x1, ... , xn - x(n-1))
xk-1 <= Xk <= xk k = 1, 2, ... , n
F '(x) dx = F(b) - F(a) (Fundamental Theorem for integrals of derivatives)
a f(x) dx = a f(x) dx (if a is constant)
f(x) + g(x) dx = f(x) dx + g(x) dx
\[ \int_a^b f(x) dx = \left[ \int f(x) \, dx \right]_a^b \]
f(x) dx + 
f(x) dx = 
f(x) dx
![\[ \int f(u) \frac{du}{dx} \, dx = \int f(u) du \]](/math/tex/062a599d472e0da223d177c460f35662.png "\[ \int f(u) \frac{du}{dx} \, dx = \int f(u) du \]")
(integration by substitution)
