Evaluate the integral.
$ \displaystyle \int_0^\pi \cos^4 (2t) dt $
this problem is from Chapter seven section to Problem number nine in the book Calculus Early. Transcendental lt's a Tradition by James Door Here we have a definite in a rural from syrup I co signed to the fourth Power of Tootie. Since we haven't even power on the coastline and there's no science present, it's going to be best to use this identity from trigonometry. Coastline Square eggs is one plus co. Sign of two eggs, all divided by two. Further says you have a coastline to the fourth power. Let's rewrite this Interbrand as coastline to the four power of two tea. We can rewrite this as co sign squared of two teams squared. Now we can apply that triggered a metric identity above two rewrite co sign squared of two T. So this becomes one plus co sign of two times two t so over forty, all divided by two and were squaring this expression. So let's evaluate this. We have a one plus two coastline of forty plus co sign squared of forty and the denominator get squared as well. So if the four of them up, OK, so using these facts, we can rewrite this interval. You're a pi. Let's pull out this one of her fourth onside general. Everyone plus to co sign fourteen Close co sign squared forty dt. Before we start evaluating dinner girls here we have a co sign squared. So let's apply our identity It once again to rewrite thisclose signed, squared. So before we do that licious copy and paste the first two terms. So we still have integral c R A pie one plus to co sign a forty plus. Now, once again using our earlier identity, we can rewrite coastlines. Where is one plus co sign of two times fourteen. So we have eighteen now over two dt. So now we can evaluate each of these intervals. So the integral of one with respect city simply t integral through co signed forty is too signed forty over four. So this is coming from a U substitution u equals fourteen. Then we have integral of one over two t over too. And lastly, we have integral of co sign of eighty over too. So that will become sign of eighty over two times eight. So this lass in a rural co sign of eighty was evaluated by using two u sub u equals eighty. So now we've evaluated integral will plug in pine zeroes are points and evaluate, so plugging in pie for tea First we have pie plus two over four Sign for pie plus poverty plus sign of a pie over sixteen. So this is from plugging in pilot for team and now we'LL plug in zero. And when we plug in zero for TV each of these four terms of ten zero So here we're just attracting zero. So using the unit circle once again signed a four by zero sign of a prize. Also zero. So we have one over four pi, plus by over two, three, five, two, two, which we could simplify. It is three by over eight, and that's your final answer.