Antiderivatives of Trig Functions

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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: 1/4∫cosu 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