Antiderivative of X

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Anti-derivatives is the reverse or opposite of derivatives. Here the function given is ‘x’ and the exponent to

which the variable is raised is ‘1’. The power rule is used to find the anti-derivative for any function which

contains a variable raised to an exponent.According to the power rule any function which has the variable

raised to the power ‘n’ is written as ‘xn’ has theanti-derivative = ∫xn dx= x(n+1)/ (n+1) + c.Hence the anti-

derivative of the function ‘x’ is 1x2/2.



Example 1: Find the anti-derivative of the function f(x) = x + 6x3


Here the given function is f(x) = x + 6x3

The anti-derivative of x is 1/2 * x2

Using the power rule, the anti-derivative of 6x3 has to be found.

Power rule states that anti-derivative of xn = ∫xn dx= x(n+1)/ (n+1) + c

Therefore, the anti-derivative of 6x3 is 6x4/4.

Hence F(x) = 1x2/2 + 3x4/2 + c



Example 2: Find the anti-derivative of the function f(x) = 15 – x.


Here the given function is f(x) = 15 - 3x.

The anti-derivative of x is 1/2 x2

Using the power rule, the anti-derivative of 15 has to be found.

Power rule states that anti-derivative of xn = ∫xn dx= x(n+1)/ (n+1) + c

15 can be written as 15 x0.

Therefore, the anti-derivative of 15 x0 is 15x1

Hence F(x) = 15 x -x2/2 + c



 

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