Indefinite integrals Use a change of variables or Table 5.6 to...

Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ 2 / (𝓍√4𝓍² ―1) d𝓍 , 𝓍 > ½

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the indefinite integral using a table of integrals, and we're given the integral of 3 divided by the quantity of X multiplied by the square root of the quantity of 9 X2 minus 4 in quantity DX, where X is greater than 2 divided by 3. So, our first step is to rewrite our integran into a form that we can actually use our integral tables with. So, our first step is to rewrite the denominator. So we'll have the interval of 3 divided by the quantity of X multiplied by the square root of the quantity, and here we're going to rewrite 9 X 2d as the quantity of 3 X in quantity squared. Then we'll have minus and instead of 4, we'll rewrite that as 2 squared in quantity. And then this gets multiplied by DX. And so now we're going to use U substitution, to get this into a form that we can actually use our integral tables with. So we're going to set U equal to 3 X, and so that means that DU is equal to 3D X. Well, we'll also need X in terms of you, so looking at our first expression of you is equal to 3 X. And solving for X, we'll see that X is equal to U divided by 3. So, rewriting our interval, we see that it's going to be equal to. The interval of here in the numerator, we have 3DX which is just going to be DU. And that's gonna be divided by the quantity of X, which is you divided by 3, and then that gets multiplied by the square root of the quantity of U2 minus 2 squad. And just simplifying this expression a bit, we'll see that this is going to be equal to. We can bring the factor of 1 divided by 3 in the denominator up to the numerator, so it'll just be 3, and since that's constant, we can pull it out front of the integral. So we'll have 3 multiplied by the integral of 1 divided by the quantity of U multiplied by the square root of the quantity of U2 minus 22 in quantity DU. And this integral is actually a form in our integral tables. So if we look up at our integral tables, we see that the integral. Of 1 divided by the quantity of X multiplied by the square root of the quantity of X2 minus A2 in quantity DX is equal to. One divided by A multiplied by the inverse sequent. Of the absolute value of the quantity of X divided by A in quantity plus D. So in our case, A is going to be equal to 2. So this is going to be equal to 3. Multiplied by the quantity. Of 1 divided by 2, multiplied by the inverse sequent. Of the absolute value of the quantity of U divided by 2 in quantity plus C. In quantity. So now all we need to do is substitute in 3 X for you to get our function back into a function of X, and doing so, this is going to be equal 2, and here we'll also distribute the 3. So we'll have 3 divided by 2. Multiplied by the inverse sequent. Of the absolute value of the quantity of 3 X divided by 2. In quantity plus C. So this here is the answer that we were looking for for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ 2 / (𝓍√4𝓍² ―1) d𝓍 , 𝓍 > ½

Key Concepts

Indefinite Integrals

Change of Variables

Differentiation Check

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