Anti Derivatives and their Rules

What are Anti derivatives?
Anti derivative is nothing but indefinite integral or primitive integral in calculus. If there is a function h, then the anti-derivative of this function will be a differential function, say H. The derivative of H will be equal to h.
H’ = h

The anti derivates are solved by a process called indefinite integration or anti differentiation, which is the opposite process of differentiation that finds the derivative.

Rules of Anti derivatives
The rules of anti derivatives are generally the reverse of the rules of the derivatives. We can say that the anti derivative of a function is equal to the sum of the derivative of the function and a constant, so it is just one step and simple.

Constant Rule
Consider a constant “a”, which has to be multiplied with the function g(y). The value obtained by multiplying the constant with the anti derivative of the function g(y) for all values of y will be equal to the anti derivative of the function g(y), which is calculated after multiplying the constant with the function g(y) for all values of y.

Sum Rule
If there are two functions f(y) and g(y), the anti derivative of the sum of the two functions will be equal to the sum of the anti derivative of the function f(y) and the anti derivative of the function g(y).

Difference Rule
If there are two functions f(y) and g(y), the anti derivative of the difference of the two functions will be equal to the difference of the anti derivative of the function g(y) from the anti derivative of the function f(y).

Reverse Rules
Listed below are the anti derivative rules that form the reverse of derivative rules:
• The anti derivative of cos y is given by sin y + a, where “a” is a constant. Thus we can say, the anti derivative of the derivative of y will result in y + a.
• The anti derivative of sin y is the negative of the sum of cos y and a.
• The anti derivative of the square of sec y is the sum of tan y and a.
• The anti derivative of the square of cosec y is the negative of the sum of cot y and a.
• The anti derivative of the product of sec y and tan y is given by the sum of sec y and a.
• The anti derivative of the product of cosec y and cot y is given by the negative of the sum of cosec y and a.

Power Rule
The power rule of the anti derivative is the reverse of the power rule of the derivative. The power rule of the derivative usually comprises of two steps as the power is brought in the front to be multiplied with the derivative and then the power is reduced by 1 and then it is simplified.

In case of power rule for anti derivative, the power rule comprises of the following two steps.
Step 1: The value of the power in the function is increased by 1. Say for example, g(y) = 6y^2. Then by increasing the power value by 1, the function will become g(y) = 6y^3.

Step 2: Divide the function g(y) obtained from step 1 by the new power value. In this example, it will be: (6 divided by 3) (y^3), which will result in g(y) = 2y^3.

Thus anti derivative of this example is given by 2y^3 + a, where “a” is a constant, as any anti derivative is the sum of the derivative and a constant “a”.