Problem 8 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{dt}{t^2 \sqrt{t^2 - 16}} $

Answer

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Video Transcript

here we have the anti derivative of one over T square times the square root of T squared minus sixteen. Because our denominator is of the form T squared minus a square. Let's go ahead and try. Substitution of the Forum T equals a Sikh and data. So here are Valium. A is for because a square to sixteen. So we have is forcing can't data from which we have The derivative DT is four c can data tan data. Now, before we go on to rewrite in a roll, let's just go ahead and deal with this radical. First we have t squared minus sixteen square root will become sixteen. Seek and squared minus sixteen. We can pull the sixteen outside the radical that becomes a four. We have Sikh and squared minus one. So this becomes four square root Tan square, which is fourth and data. So we have integral D t, which is given over here for sick and data tan data and the denominator. We have this t squared out here in the front. That's a sixteen seek and square data and then we just evaluated the square root. And as for Sandra was God and cancels much as we can. You see these force Cancel. Those tangents will cancel. So we're left with one over sixteen, and then you could cancel out one of these sea cans. So we have one over seek and data using the definition of Sikkim. This is because I data and inside rivet of the coastline is simply sign. So we have signed data over sixteen plus C. Now, this is the point in which we use the the right triangle in orderto rewrite our answer in terms of TV. So let's come back up over here to our tourism. So from this we have seek and data equals t over four. Those trucks are right triangle from this. So let's put they towed over here in the bottom. Right? So we have seeking is high pop news over adjacent. And then we can use put that rent the room to find the remaining side age. Solis, he and make some room. Here we have a squared. Plus four squared is Theis. Where? So H equals square root. Is he squared? Minus sixteen. Now, using this fact, now that we have all three sides of the triangle, we could find sign of data. So sign of data is H over tea. So we have square of T squared minus sixteen over tea. Don't forget about the sixteen over here. Someone's put a sixteen t plus C, and that's our answer.