Problem

Integrate using the method of trigonometric subst…

08:43

Problem 142 Medium Difficulty

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.
$$\int \frac{\sqrt{x^{2}-25}}{x} d x$$

Answer

$5 \sqrt{x^{2}-1}-5 \sec ^{-1}\left(\frac{x}{5}\right)+C$

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus

Chapter 3

Techniques of Integration

Section 3

Trigonometric Substitution

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Video Transcript

Okay, so we have the integral of the square root of X squared minus 25 over X, the X Let's draw out or triangle. So we have explored minus 25. So we know that our hypothesis is that the ex one of our sides is gonna be five. And the other side would be the square root of X squared minus five. All right, 25. Okay. Now let's let's or x equal to five, seeking dinner. So then, are the X is that that's by sequins, Data, tangents, data. You did it Didn't make it. So if we have ness Mmm. Hey, Should see Dixie. Mmm. Okay, So let's figure out what tangents of data is. And you know, Veda is fine, but co sign. So we have this group of X squared minus 25 over five. So we have five. Tensions of data is equal status, girl roots of X squared minus 25. So this is what we want over here. You know, X is equal to five seconds later. So it's plugging what we know. Get the integral of five tangents. Data over five. She can't get up. Times are the X, which is five sequins data. And instead he did up. Okay, One of our fires cancel. Are she can also cancels five engines squared data data. Okay, we know that, but Chandon score data. That's the sequel to seek inns squared theta minus one D data. Okay, they're looking to read this. Get five times tangents of, But it's five. I see. Well, what is tangent of data? We said tagged inhibitor is this thing over here? So we have five cents a square root of X squared minus 25 over five minus five. Data and data is equal to Let's see X. You don't have to hear this. That's X over five and speaking in verse and the politics feeling of that five times. Thanks. Inverse of X over five. Let's see, kid. Oh, it is. She don't have a five year. They put out the five. It's like this. So the solution is actually score boots up X squared minus 25 over five minus. Oh, not over fives. Because if I pull in our five cancels minus five seconds, inverse of X over five plus

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