Integrate cos^2x

To integrate cos^2x, also written as ∫cos²x dx, cos squared x, and (cos x)^2, we start by using standard trig identities to simplify the integral.

We start by using the Pythagorean trig identity sin²x+cos²x=1.

We rearrange it for cos²x.

We recall the double-angle trig idnetity shown above.

We arrange this identity for sin²x, so that we can substitute it into the first trig identity.

The contents of the brackets is what we have substituted into the first trig identity.

We open out the brackets and pay attention to the sign change.

The cos²x goes to the LHS with a sign change. By grouping them together, we can add them to simplify.

In this step, we add the cos²x terms.

We divide throughout by ½ so that we have an expression for cos²x.

As you can see, we now have a new integral which is the same as the original but easier to perform. The constant ½ is very simple to integrate, and now we must endeavour to focus our attention on integrating the second term cos2x.

In order to integrate cos2x we use the u substitution method.

Let u=2x

Therefore, the differential is 2.

We rearrange for dx as shown above.

As you can see, we now have an equivalent integral in terms of u that is simple to perform. As usual, we tuck away the constant ½ outside of the integral.

At this step, the integration is simple, as you can imagine.

We now reintroduce the constant ½ to the result.

Since we know that u=2x, we can substitute that back in to give the above result.

Coming back to this expression, which we made earlier, we are now in a position to substitute the answer for the integral of cos2x.

The contents of the brackets is the part we have substituted, and C is the integration constant.

We now simplify the answer by removing the brackets.