Antiderivative

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The method of finding the Antiderivative of a function is also known as the method of Integration. There are

two types of antiderivatives, one being the indefinite integrals where the constant ’c’ is includedin the answer

of the function. The other type of antiderivatives is the definite integrals where the constant ‘c’ is not included

and the final solution of the antiderivative is computed by substitution of numbers.



Example 1: Find the anti-derivative of the function, f(x) = 8x3- 10x + 9


The Power Rule of Integration says that ∫ (x) n dx = x (n+1)/ (n+1) + c

where ‘c’ is a constant

Using the above formula we get,

∫ f(x) dx = 8 * x3+1/ (3+1) – 10 * x1+1/ (1+1) + 9x + c

∫ f(x) dx = 8 * x4/ 4 – 10 * x2/ (2) + 9x + c

∫ f(x) dx = 2x4– 5x2 + 9x + c


 
Example 2: Find the antiderivative of the definite integral value of the function, f(x) = 3x2 + 2x and

‘x’ ranging from 0 to 2.


 ∫xn dx= x(n+1)/ (n+1)


Apply the above formula for the given function, we get

∫(fx) dx = 3* x2+1/(2 + 1) + 2*x1+1/(1 + 1)

∫f(x)dx = x3 + x2

First substitute x =0 and x= 2 in the above answer.

When x=0, ∫f(x) dx= 03 +02= 0

When x=2, ∫f(x)dx= 23 + 22 = 12

Now subtract 12 - 0 = 12

Hence the antiderivative of given f(x) is 12.

 

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