Antiderivative of Trig Functions

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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: 1/3∫sinu 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: 1/2∫secu tanu du = 1/2(secu) + c = 1/2(sec2x) + c

Hence ∫sec2x tan2x dx = 1/2(sec2x) + c