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.
37. ∫ dx / √(x² + 10x), x >

37. ∫ dx / √(x² + 10x), x >

Accepted Answer

Hello. In this video, we are going to be evaluating the indefinite integral. We are given the integral DX divided by the square root of X2 + 8X. Now, in order to approach this integral, let's go ahead and simplify what is inside the square root function. So, what we are given inside the square root function is X2 + 8X. Now, we can simplify this expression by completing the square. If we were to complete the square, we get X2 + 8 X plus 16 and in order to balance the equation, we will have to subtract 16. Now, after completing the square, if we were to group the three terms together, we can factor these three terms to be X + 4 quantity squared minus 16. Now, if we were to rewrite the integrand using the simplification, we can rewrite the given integral to be the integral of one divided by the square root of the quantity X + 4 quantity squared minus 16 DX. Now, we can further simplify this by applying a U substitution. We are going to allow you to equal to X + 4, and that means that DU is going to equal to DX. So, by using a U substitution, we can further simplify this integral to be the integral of 1 divided by the square root of U2 minus 16DU. Now this type of integral does have an identity with it. It matches the following integral. If we had the integral of 1 divided by the square root of X2 minus aqua quantity, let's say A2d, then this is equal to the natural logarithm of X plus the square root of X2 minus A2, and because it is the indefinite integral, we will add on a general constant C. Now, in our integral here, our squared quantity is 16, so A2 is equal to 16 in our case, and by applying this identity, we can rewrite or solve the integral to be the natural logarithm of U plus the square root of U squared. Minus are defined A2, which is 16 plus C. And now what we need to do is we just need to resubstitute you into terms of X. That is going to leave us with the natural logarithm of X + 4. Plus the square root of X + 4 quantity squared minus 16. Plus C And this is going to be the simplest form of the indefinite integral that we were solving for, and that means that the overall solution is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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.
37. ∫ dx / √(x² + 10x), x >

Key Concepts

Indefinite Integrals

Completing the Square

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.37. ∫ dx / √(x² + 10x), x >

Key Concepts

Indefinite Integrals

Completing the Square

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.
37. ∫ dx / √(x² + 10x), x >