Question

Evaluate the integral: ∫cos⁴ x dx

Answer 1

In the given question we need to find the integral of the cosine function raised to power four which is cos⁴x and also we will make use of the fact that the integration of cosx is sinx and 1dx is x.

Now, in order to integrate the given cosine function what we need to do is use the cosine identity as follows:

\(\cos^2x = \frac{1 + \cos 2x}2\)

So, now we need to integrate ∫cos⁴x dx and applying the identity we get,

Now, applying the identity again and integrating using the fact that integration of cosx is sinx and 1dx is x we get,

Now, simplifying it further we get,

\(\frac 14 \left(\sin 2x+ \frac 32 x+\frac{\sin 4x}8\right)\)

Therefore, the integral of the given cosine function cos⁴x dx we get

\(\frac 14 \left(\sin 2x+ \frac 32 x+\frac{\sin 4x}8\right)\)

Answer 2

It is given that