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# Evaluate the integral.$\displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1 + 4 \cot x}{4 - \cot x}\ dx$

## $\ln \left(\frac{4 \sqrt{2}}{3}\right)$

#### Topics

Integration Techniques

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### Video Transcript

Let's start off here by just rewriting co tangent. So using the definition of cool attention, we can write this as co sign eggs over Sine X and then on the bottom. And then let's go ahead and simplify these fractions and we can breathe right. This is sine X plus four co sign and then you have four sign minus co sign after cancelling out those signs. This is what we have now. From this point, let's go ahead and try Yusa, take you to be that denominator. Then we know that, do you? Dx And that's precisely what we see over here in the numerator four Co Zion plus I the ex. Also, we should watch out for these limited generation. Those will change. So to find out there you values go ahead and plug in these X values into the use of So if X equals power for so just go right ahead. Plug in pie before in for X for room two over too. Bring two over too. So three radical to over too. That's our new lower limit. And for the upper limit, go ahead and plug in pirate too, and Rex and there you'LL get four minus zero equals four. So we can rewrite this three root, too over to before, and then we just have one over you. Do you recall the D'You accounts for this over here and then the yuan The denominator corresponds to this two. It's evaluate that in a rule that's natural, odd, absolute value, you then go ahead and plug in three and room to over two and four and then just subtract these end points and simplify and you'LL get it. Four article, too over three inside the longer them and that's your final answer.