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# Antiderivatives

In calculus, an antiderivative, primeval or unclear integral of a function f is a task F whose imitative is equal to f, i.e., F? = f. The process of solving for antiderivatives is called antidifferentiation and its differing occupation is called differentiation, which is the development of result a derived. The antiderivative, or primeval, of a intricate-valued function g is a function whose intricate derivative is g. Antiderivatives are connected to definite integrals through the basic theorem of calculus the exact integral of a task over an interval is different to the difference between the values of an antiderivative evaluate at the endpoints of the distance.

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## Antiderivatives Conditions

• The basic of a satisfactory, state for a function f to have an antiderivative is that f has the middle value property. That is, if [a, b] is a subinterval of the sphere of f and d is any actual number between f (a) and f (b), then f(c) = d for some c between a and b.

• The set of discontinuities of f should be a meager set. They can make some significance values of f having an anti-derivative, which has to locate as it’s location of discontinuity.

• If f has an antiderivative, is bound on close finite subinterval of the domain and has a set of discontinuities of Lévesque measure 0, then an antiderivative may be found by integration.

The method of finding antiderivatives is called antidifferentiation or integration:

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

=>   int  f(x) dx = F(x) + C.

The expression F(x) + C is the general antiderivative of ƒ. If ƒ and g are defined on the same interval, then the general antiderivative of the sum of ƒ and g equals the sum of the general antiderivatives of ƒ and g.

## Antiderivative Formulas

int xn dx = (x^(n + 1))/(n + 1) + C  (for n ? - 1)

int 1/x dx = ln |x| + C

int ex dx = ex + C

int sin x dx = - cos x + C

int cos x dx = sin x + C

int sec2 x dx = tan x + C

int cosec2 x dx = - cot x + C

int sec x tan x dx = sec x + C

int cosec x cot x dx = - cosec x + C

int tan x dx = ln |sec x| + C

int cot x dx = - ln |cosec x| + C

int sec x dx = ln |sec x + tan x| + C

int cosec x dx = ln |cosec x - cot x| + C

## Finding Antiderivative

Below are the examples on antiderivatives -

Example 1: Find antiderivative for the function, f(x) = 3x2 + 7x.

Solution:

Step 1: Given function

f(x) = 3x2 + 7x

int f(x) dx = int (3x2 + 7x) dx

Step 2: Separate the integral function

int(3x2 + 7x) dx = int 3x2 dx + int 7x dx

Step 3: Integrate each function with respect to ' x',

int(3x2 + 7x) dx = (3x^3)/3 + (7x^2)/2 + C

= x3+ (7x^2)/2 + C

Example 2: Find antiderivative for the function, f(x) = 2/x^7

Solution:

Step 1: Given function

f(x) = 2/x^7

int f(x) dx = int 2/x^7 dx

Step 2: Integrate the function 2/x^7 with respect to ' x ',

int 2/x^7 dx = int 2x-7 dx

= 2((x^(-7 + 1))/(-7 + 1))

= 2 ((x^-6)/(-6)) + C

= - 1/(3x^6) + C

Example 3: Find antiderivative of y = 2cos 8x

Solution:

Step 1: Given function

y = 2cos 8x

int y dx = int 2cos 8x dx

Step 2: Integrate the given function y = 2cos 8x with respect to ' x',

int2cos 8x dx = 2sin (8x) 1/8

= (sin (8x))/4

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