7–84. Evaluate the following integrals.
25. ∫ [1 / (x√(1...

Question

7–84. Evaluate the following integrals.

25. ∫ [1 / (x√(1 - x²))] dx

Accepted Answer

Hello. In this video, we are going to be evaluating the following integral. We are giving the integral of 1 divided by X multiplied by 4 minus X squad. Now, we can approach this integral by using trigonometric substitution. This integral is in a form where we can allow X to equal to a multiplied by sine of theta. Now, in this case here, A is going to be the square constant that is within the square root term. Here we have A2 is equal to 4, so our A term is going to equal to 2. And so our trigonometric substitution is going to be that X is equal to 2 multiplied by sine theta. Now, since we are substituting X into terms of theta, we need to find a substitution of DX in terms of theta as well. By taking the derivative of both sides of the substitution, we get DX is equal to 2 multiplied by cosine of theta D theta. So, if we were to apply these substitutions, we can rewrite the integral as the following. We will have the integral of 1 divided by X, which is going to be 2 sin of theta. Multiplied by the square root of 4 minus X2, which is going to be 4 square theta. Then we are going to multiply this by 2 cosine of theta D theta. All right, there is a bit of simplification we can do here. First, we're going to multiply the numerators together, so we're going to have two cosine of theta in the numerator. Furthermore, we can factor out a 4 from the square root, and that is going to leave us with 2 sin of theta multiplied by 2 multiplied by the square root of 1 minus cosine sine square theta. Now, since we have a factor of 2 in both the numerator and denominator, we can eliminate these factors. 1 minus square theta can be rewritten as cosine squared theta, and that means that if we were to pull the two from the denominator to the front of the integral, we will have one half multiplied by the integral of cosine theta, divided by sine theta, cosine theta D theta. Now, we have cosine in both the numerator and denominator, so we can reduce those terms. And if we were to rewrite one divided by sine theta as cosequent theta, we can rewrite the integral to be one half multiplied by the integral of cosequent theta D theta. Now, we can solve this integral by using the table of integrals. The identity for co-seeking of theta is defined as the natural logarithm multiplied by tangent theta divided by 2, which is a half angle for for tangent. And so that means that this integral is going to simplify to be one half multiplied by the natural logarithm. Of the half angle tangent data divided by 2, and because we were working with the indefinite integral, we'll add a generic constant C. Now, we don't want our answer to have any half angle values, so let's go ahead and apply the half-angle identity for tangent. That will allow us to rewrite the solution to be one half multiplied by the natural logarithm of 1 minus cosine of theta, divided by sin of theta plus C. Now, now that this is the simplest form of the solution, we have to convert our solution from terms of theta back into terms of X. Now recall that our original X substitution was that X is equal to 2 sin of theta. We can create a right triangle using this statement in order to rewrite cosine data and sine theta back in the terms of X. Now, if we were to divide both sides of this equation by 2, we get sin of theta is equal to X divided by 2, and that means that we can create a right triangle with respect to this value of sine. This is going to give us the opposite value, which is of theta, which is going to be X, and we're going to have a hypotenuse of 2. Now let's go ahead and solve for the missing side. By Pythagorean identity that is going to give us 2 squad, which is 4 equal to our missing side B2 plus X2. By subtracting and taking the square roots, the missing side is going to equal to the square root of 4 minus X2. And so now all we need to do is we just need to write the cosine value for this right triangle. Now, cosine is going to be the adjacent side of theta divided by the hypotenuse, and that is going to be the square root of 4 minus X2d divided by 2. And so now if we were to plug in these values into our solution, we can rewrite the solution to be one half multiplied by the natural logarithm of 1 minus cosine, which is the square root of 4 minus X2 divided by 2, and this will be divided by sine, which is X divided by 2 plus C. And by further simplifying, this is going to allow us to get a final solution of one half multiplied by the natural logarithm of 2 minus the square root, 4 minus X2. Divided by X plus C, and this is going to be the final solution to this problem, and the overall solution is going to be A. 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.

7–84. Evaluate the following integrals.
25. ∫ [1 / (x√(1 - x²))] dx

Key Concepts

Integration

Substitution Method

Trigonometric Substitution

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