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Problem 17 Medium Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int y \sqrt{6 + 4y -4y^2}\ dy $

Answer

$$\frac{1}{24}\left(8 y^{2}-2 y-15\right) \sqrt{6+4 y-4 y^{2}}+\frac{7}{8} \sin ^{-1}\left(\frac{2 y-1}{\sqrt{7}}\right)+C$$

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Video Transcript

Okay, so this question wants us to integrate this function using a table. So to do this, we need to get this into one of our forms that we can use with our table. So most of our forms have something squared inside the square root. So let's see what we can D'oh! To factor this expression. So if we factor this we see, we have ah four y squared with the negative in front and then a four. Why? So we sort of complete the square, so to speak. We can put a negative sign in front and get four. Why squared minus for why? And then Well, what completes this square? Well, just one. So then we have a negative one there. So negative one. Plus what has two equal six? Well, seven. So now we can simplify this expression as the integral of why times the square root of seven minus the complete square expression to why minus one all squared. Okay, so now we see what our choice of you is going to be. So let's let you equal to y minus one. So that means that do you equals to do y or do you over two equals D y. But then we see we also have just a regular. Why in our expression, So we should solve for that. So why is equal to you plus one all divided by two. So plugging back in we get the integral of you plus one all over too. Replacing why times the square root of seven minus You squared times two d u Sorry, Do you over too. So then this simplifies toe 1/4 times the integral of you plus one. Outside times the square root of seven minus you squared, do you? So now from here this still doesn't really look like something we can evaluate. But in actuality, if we split this into two different inside girls, we can factor it using both tails. So we get 1/4 time's the integral of while the you term plus 1/4 times plus one term. And now we can evaluate each of these expressions separately to give us a final answer of 1/4 times evaluating each of these. We get you over too. Times the square root of seven minus you squared plus seven over to sign in verse of you over squared of seven plus sorry, minus minus one Over three times you squared seven minus You squared to the three halves plus c And then if we back substitute everything in We get remembering that seven minus you squared turns into our original expression. So this is to why minus one looking back in for you Times are square root and we'll just copy down the original one plus r sine inverse terms 7/8 sine inverse of you over too. We're sorry. Not you over too. We're plugging back in to y minus one or scared of seven. Plus, uh, minus 1/12 time's original expression all to the three halves, plus our integration Constant c. So this is a very big anti derivative. But again, that just came from the fact that we had to split this into two separate answer girls and evaluated that way.

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