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What is the integral of cos22x?

Answer
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Hint: We will use the trigonometric identities to find the integral. First, we will change the given trigonometric function to an equivalent simple form. And then, we will integrate the function with respect to x. We will use the identity cos2x=1+cos2x2.

Complete step by step solution:
Let us consider the given trigonometric function cos22x.
We are asked to find the integral of the given function with respect to x.
That is, we need to find the value of cos2xdx.
We know the trigonometric identity given by cos2x=1+cos2x2.
In this identity, we will put 2x instead of x, since our function contains 2x.
We will get cos2(2x)=1+cos2(2x)2.
We can write this as cos22x=1+cos4x2.
Now, we can change the integral using the above equation.
We will get cos22xdx=1+cos4x2dx.
Let us take 12 out of the integral sign, since it is a constant term. We know that we can take the constant term if it is multiplied with a variable term. If the operation is addition, then we cannot take the term out.
Now, we will get cos22x=12(1+cos4x)dx.
Now, we can use the linearity property of integration.
We will get cos22x=12{1dx+cos4xdx}.
Let us consider the first integral, 1dx=x.
We know that the integral of the Cosine function is the Sine function.
So, from the second integral, we will get cos4xdx=sin4x4.
So, the integral will become cos22xdx=12{x+sin4x4}+C.
Hence the integral of the given function is cos22xdx=x2+sin4x8+C.

Note: Remember the similar trigonometric identity sin2x=1cos2x2. By applying this identity, we can find the integral of sin22x in the exact way we have found the integral of cos22x. The linearity property of integration is (af(x)+bg(x))dx=af(x)dx+bg(x)dx where a,b are constants and f,g are functions of x.

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