Antiderivative of Trig Functions

Antiderivative of Trig Functions Antiderivative is the method of finding the area covered by a function when graphed on a coordinate plane. Antiderivative is the opposite method of thederivative method of a function and hence the name. Antiderivative of trig functions is the method of finding the integral of the trigonometric functions which include functions like sinx, cosx, tanx, etc. Example 1: Find the antiderivative of the trigonometric function cos4x. The antiderivative notation of the given trigonometric function is: cos4x dx We can use u-substitution method to find its antiderivative. Let u = 4x, then du = 4dx, dx = du/4 Now substitute the above u value in the given function We get, cos4x dx = cosu * du/4 = 1/4 cosu du Formula for antiderivative of cosx = cosxdx = sinx + c Applying the above formula, we get: 1/4cosu du= 1/4(sinu) + c = 1/4(sin4x) + c Hence cos4x dx = 1/4(sin4x) + c Example 2: Find the antiderivative of the trigonometric function 2sinxcosx The antiderivative notation of the given trigonometric function is: 2sinx cosx dx We can use u-substitution method to find its antiderivative. Let u = sinx, then du = cosxdx, dx = du/cosx Now substitute the above u value in the given function We get, 2sinx cosxdx = 2 u cosx * du/cosx Cancelling cosx up and down we get: 2 u du Formula for antiderivative of x = x dx = x2/2 + c Applying the above formula, we get: 2 udu = 2(u2/2) + c = u2 + c Hence 2sinx cosx dx = sin2x + c