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`(2a^(2))/(3)``(a^(2))/(3)``(8a^(2))/(3)``(16sqrt2a^(2))/(3)`

**Fundamental Theorem of Definite Integration**

**`int_a ^b f(x) dx = phi(b) - phi(a)`**

**Examples: `int_2 ^4 x / (x^2 + 1) dx`**

**Definite integration by substitution**

**Examples: `int_0 ^1 sin^-1 ((2x )/ (1 + x^2)) dx`**

**Property 1: Integration is independent of the change of variable. `int_a ^b f(x) dx = int_a ^b f(t) dt`**

**Property 2: If the limits of a definite integral are interchanged then its value changes. `int_a ^b f(x) dx = - int_b ^a f(x) dx`**

**Property 3: `int_a ^b f(x) dx = int_a ^c f(x)dx + int_c ^b f(x) dx`**

**Property 4: If `f(x)` is a continuous function on `[a,b]` then `int_a ^b f(x) dx = int_a ^b f(a+b-x) dx`**

**Property 5: If `f(x)` is a continuous function defined on `[0,a]` then `int_0 ^a f(x) dx = int_0^a f(a-x) dx`**