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Let `f(x) ` is continuous on `[a, b].` If `G(x) ` is permanent on` [a, b]` and `G'(x) =f(x)` for all` x in (a, b),` then `G` is called an antiderivative of `f.` We know how to put up antiderivatives by integrating. The function `F(x) =int ^x_a f (t) dt` is an antiderivatives for` f` because it can be shown that` F(x)` constructed in this way is continuous on `[a, b] ` and `F, (x) =f(x) ` for all `x in` `(a, b).`

Let` F(x)` is any antiderivative for` f(x).`

• For any constant `C, F(x) + C` is an antiderivative for `f(x).`

**Proof:**

Since `(d)/(dx)[F(x)]=f(x),`

`(d)/ (dx) [F(x) + C]`

`= (d)/ (dx) [F(x)] + (d)/ (dx) [C]`

`=f(x) +0`

`=f(x)`

Hence `F(x) + C ` are an antiderivative for `f(x)` .

• Every antiderivative of` f(x) ` can be written in the form

`F(x) + C`

for some `C.` That is, each two antiderivatives of f vary by at most a constant.

**Proof:**

Let `F(x)` and `G(x) ` are antiderivatives of `f(x)` . Then `F'(x) =G'(x) =f(x), ` so `F(x)` and G(x) differ by at the majority an invariable.

The development of finding antiderivatives is called antidifferentiation or integration:

`(d)/(dx)[F(x)] = f(x) lArr rArr int f(x)dx=F(x)+ C .`

`(d)/ (dx) [g(x)] = g' (x) lArr rArr int g' (x) dx = g(x) + C.`

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• `(d)/ (dx) [int f(x) dx] = f(x).`

**Proof:**

Let `int f(x) dx =F(x) ` , where` F(x)` is an antiderivative of `f` . Then

`(d)/ (dx) [int f(x) dx] = (d)/ (dx) F(x) = f(x)`

` int [alpha f(x) + beta g(x)] dx = alpha int f(x) dx + beta int g(x)dx .`

**Proof:**

We need only show that alpha int `f(x) dx + beta int g(x) dx ` is an antiderivative of int` [alpha f(x) + beta g(x)] dx:`

`(d)/ (dx) [alpha int f(x) dx + beta int g(x) dx]`

`= alpha (d)/ (dx) [int f(x) dx] + beta (d)/ (dx) [int g(x) dx]`

`=alpha f(x) +beta g(x)`

**Examples**

1. Every antiderivative of `x^2 has the form x^3/3 + C`

since `(d)/ (dx) [(x^3)/ (3)] =x^2.`

2. `(d)/ (dx) [intx^5 dx] = x^5.`

If `G(x) ` is continuous on `[a, b]` and `G'(x) =f(x) ` for all x in `(a, b),` then `G` is called an antiderivative of `f` .

We know how to make antiderivatives by integrating. The function `F(x) = int ^a_x f (t) ` dt is an antiderivative for `f.` In detail every antiderivative of f(x) can be written in the form `F(x) + C` , for some `C`.

`(d)/ (dx) [F(x)] =f(x) lArr rArr int f(x) dx = F(x) + C.`

`(d)/ (dx) [g(x)] =g'(x) lArr rArr int g'(x) dx = g(x) + C.`