Antiderivatives of Sin and Cos Functions

Antiderivatives or indefinite integrals can be found out for all the trigonometric functions. Antiderivative of tanx, cosx and sinx can be easily calculated. In this article, let us discuss about the anti derivatives of various cos and sin functions.

Antiderivative of Cos Function

Antiderivatives can be calculated for various trigonometric cos functions easily using the identities. Listed below are the anti derivatives of various cos functions:

·         The anti derivative of cos y is given by the summation of sin y and the constant a.

·         The anti derivative of the function square of cos y can be calculated easily by using the identity cos^2(y) = ½(1+ cos (2y)). By applying this identity to calculate the anti derivative of the function cos^2(y), we get indefinite integral of ½(1+ cos (2y)), which results in half of y added with quarter of sin 2y and the constant a.

·         The anti derivative of cos ny i.e. the indefinite integral of cos ny dy is given by 1/n multiplied with sin ny and added to the constant a.

Antiderivative of Sin Function

Listed below are some anti derivatives of the trigonometric sin y function used with a constant “a”:

·         The antiderivative of the function square of sin y is calculated by using the identity, sin^2y = ½(1- cos (2y)). By applying the identity, we rewrite the anti derivative of the function sin^2y as ½ multiplied with the anti derivative or indefinite integral of (1- cos (2y)) which results in (1/4) sin2y subtracted from (1/2) y and then added with the constant a.

·         The antiderivative of cube of sin y is given by 1/3 multiplied with cube of cos y, which is subtracted by cos y and added to the constant a.  The anti derivative or indefinite integral of sin^3 y dy can be rewritten as the integration of (1-cos^2 y) multiplied with sin y dy.  If we consider cos y as t then dt is given by the negative of sin y dy. By applying this, the equation can be rewritten as the integration of (1-t^2)(-dt) which results in t^3/3 subtracted by t and added to the constant a. By using the values of t and dt in the equation, we get the anti derivative of sin^3 y as (1/3)cos^3y subtracted by cos y and added to the constant a.

·         The antiderivative of sin ny is given by the negative of 1/n multiplied with cos ny added with the constant a.

Antiderivative of Cos and Sin function

The antiderivative of the function cube of cos y multiplied with square of sin y is given by 1/5 of sin^5(y) subtracted from 1/3 multiplied with cube of sin y and then added with the constant a.