Use the table of integrals at the back of the text to evaluate...

Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ dx / (x √(x + 4))

Accepted Answer

Welcome back, everyone. Using a standard integral table, evaluated the following indefinite integral. Integral of DX divided by X square root of X + 9. For this problem, let's begin with substitution. Let's suppose that U is equal to square root of X + 9. What we can do now is solve for X. So, let's go ahead and square both sides. We can show that the U squared is equal to X + 9, and therefore, X is equal to U 2 minus 9, right? Now, let's go ahead and identify the X divided by DU. That's the derivative of X with respect to you, and we are going to get to you. So now if we solve for the X, the X is equal to 2 you multiplied by the U. And now we can go ahead and rewrite our integral. Let's label it as I. In terms of UN DU. So in the numerator, we have DX which is 2 you multiplied by DU. In the denominator, we have X, which is EU 2 minus 9 multiplied by square root of X plus 9, which is EU. Notice that we can cancel out you and we can factor out to from the integral. So we're going to get to integral, DU. Divided by EU square minus 9. And now we can use the table of integrals to evaluate that. Let's remember that integral of the EU. Divided by U2 minus A2 is going to be equal to 1. Divided by 2A multiplied by LN of the absolute value of U minus A, divided by U plus A. Plus a constant of integration C where A is greater than 0. So this is the formula that we can use from standard integral tables as well. And then we can show that our integral I is going to be equal to. First of all, we have 2, right? M multiplied by. By analogy, we're going to take 1 divided by 2A. Now what is A? Well, essentially looking at our form, we can see that A2d is equal to 9 because we have U2 minus 9, which is U2 minus A2. So A2 is 9, meaning A is equal to 3. So in this case, all that we have to do is use A equals 3 to get 1 divided by 2 multiplied by 3. Multiplied by a land of the absolute value of You Which is square root of x + 9 minus A, so we're subtracting 3. Divided by U + A, so that's square root of X + 9 + 3. Plus a constant of integration C. Notice that now we can cancel out the 2 and we simply get a 1/3 in front of the integral, right. I'm sorry, in front of the natural logarithm. So that our final answer is. 13 natural log of the absolute value of square root of X + 9 minus 3 divided by square root of X + 9 + 3 + a constant of integration C. That's our final answer and thank you for watching.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ dx / (x √(x + 4))

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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(x + 4))