7–40. Table look-up integrals Use a table of integrals to eval...

Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
31. ∫ √(x² - 8x) dx, x > 8

31. ∫ √(x² - 8x) dx, x > 8

Accepted Answer

Welcome back, everyone. Evaluate the indefinite integral integral of square root of X2 minus 10 XD X where X is greater than 10 using a table of integrals. For this problem we're going to use one of the identities, which is integral of square root of u2 minus a 2Du where A is a constant is equal to. You Divided by 2 Square root Of u2 minus A2. Minus A square divided by 2. Multiplied by LN of the absolute value of U plus square root of u2 minus a2 plus a constant of integration C. So now, what we're going to do is understand that we have X2 minus 10 X each term under the radical contains X. So what we can do is simply consider X2 minus 10 X and complete the square. So that would be X. Minus 10 divided by 2 is 5. We squared the whole term, right? And we have to understand that at the end for this expression, we would add 25, so to ensure that we get X2 minus 10 X on each side, we have to subtract 25 on the right. Because let's recall that X minus 5 squad is going to be X2 minus 2 multiplied by X multiplied by 5 plus 5 squad, right? So we're subtracting 5 squad, and we can rewrite it as 5 squad. Well then Now, what we can show is that X minus 5 is our EU, right? And 5 is our A. So we got U2 minus A2d. And now what we're going to do is identify the you divided by DX because we're introducing Substitution You equals. X minus 5. So the derivative of U with respect to X is 1, right? The derivative of X minus 5 is 1. And it follows that the U is equal to D X. So that we can rewrite our integral as square root of. You squared. Because originally we had square root of X minus 5 square minus 5 square DX. So in terms of you, that would be square root of. U 2 minus 52 DU because the X is DU. And now we are ready to evaluate, right? Let's understand that our AS 5. So we begin with you divided by 2. Let's recall that U is X minus 5, so we get X minus 5 divided by 2. Square root of. U 2 minus a 2. Let's recall that this is X2 minus 10 X. Because essentially we have shown that X minus 52 minus 52 this X2 minus 10 X, right? So essentially we can write its simplified form, which is square root of X2 minus 10 X. And now we are subtracting A2, which is 5 squad, and that's 25 divided by 2. LM of the absolute value of U. Which is X minus 5 plus square root of U2 minus A2, which is once again X2 minus 10 X. Plus the constant of integration C. But now we can drop the absolute value for the natural logarithm because X is greater than some. So for any X, X minus 5 plus the radical term is going to be positive, right? Essentially, X minus 5 is going to be greater than 0 for sure, and the radical term is always positive. Well done, so we have our final answer and thank you for watching.

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
31. ∫ √(x² - 8x) dx, x > 8

Key Concepts

Completing the Square

Indefinite Integrals

Integral Tables

Watch next

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.31. ∫ √(x² - 8x) dx, x > 8

Key Concepts

Completing the Square

Indefinite Integrals

Integral Tables

Watch next

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
31. ∫ √(x² - 8x) dx, x > 8