Antiderivative of Trig Functions

Antiderivative of Trig Functions Antiderivative of a function is the method of finding integral of a given function. In this method we use different rules like Power rule, Substitution rues etc. and the antiderivative of the function is calculated. Antiderivative of trig functions is finding the integral of any trigonometric function. Different techniques are used in order to get the solution of antiderivative of the trigonometric functions. Example 1: Find the antiderivative of the trigonometric function sin3x. The antiderivative notation of the given trigonometric function is: sin3x dx We can use u-substitution method to find its antiderivative. Let u = 3x, then du = 3dx, dx = du/3 Now substitute the above u value in the given function We get, sin3x dx = sinu * du/3 = 1/3 sinu du Formula for antiderivative of sinx = sinxdx = -cosx + c Applying the above formula, we get: 1/3sinu du = 1/3(-cosu) + c = -1/3(cos3x) + c Hence sin3x dx = -1/3(cos3x) + c Example 2: Find the antiderivative of the trigonometric function sec2xtan2x. The antiderivative notation of the given trigonometric function is: sec2x tan2x dx We can use u-substitution method to find its antiderivative. Let u = 2x, then du = 2dx, dx = du/2 Now substitute the above u value in the given function We get, sec2x tan2x dx = secutanu * du/2 = 1/2 secutanu du Formula for antiderivative of secx * tanx = (secx)(tanx)dx = secx + c Applying the above formula, we get: 1/2secu tanu du = 1/2(secu) + c = 1/2(sec2x) + c Hence sec2x tan2x dx = 1/2(sec2x) + c