7-56. Trigonometric substitutions Evaluate the following integ...

Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

42. ∫ 1/(x²√(9x² - 1)) dx, x > 1/3

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the interval using trigonometric substitution, and we're given the interval of 1 divided by the quantity of X2 multiplied by the square root of the quantity of 16 X2 minus 4 in quantity DX, where X is greater than 1/2. So we're given the interval of 1 divided by the quantity of X2 multiplied by the square root of the quantity of 16 X2 minus 4 in quantity d X, and we want to evaluate this integral using Trigonometric substitution. So, we're going to set X equal to 1/2 multiplied by Deant of data. And that means that DX is going to be equal to 1/2 multiplied by deacon of theta, multiplied by tangent of data. The theater So, rewriting our interval using substitution, this interval is going to be equal to. The interval Of 1/2 multiplied by equant of theta multiplied by tangent of theta. Divided by the quantity, and here we have the quantity of 1/2 multiplied by equant of theta in quantity squared multiplied by the square root. Of the quantity, and here we'll have 16 multiplied by 1/2 squared, which will give me 4, and then that gets multiplied by 2 squared. Data and then we'll have -4 in quantity. The Theta. But now we want to simplify this expression. So the factor of 12 will be divided by a factor of 1/4 in the denominator, which could be an overall factor of 2, which we can pull out in front of the interval. So this is going to be equal to 2 multiplied by the integral and in the numerator. One factor of sequence of data will cancel out with one factor of sequence of data in the denominator. So in the numerator, I just have a tangent of data. Then this gets divided by. The quantity of 20th of data. Multiplied by, and here we can factor out a 4 out of each term inside the square root, and we take the square root of 4. That's going to give us 2, so we'll multiply that by 2, and that gets multiplied by the square root of the quantity of squared the minus 1 in quantity, and then we have the theta. And so simplifying this further, the factor of 2 in front of the interval will cancel out with a factor of 2 in the denominator, which will make it 1. So this is just gonna be equal to the interval of tangent of data. Divided by the quantity of equant of data. And here we're going to have taken squared data minus 1, which will give us tangent squared, and we're taking the square root of that, so we'll just have the square root of tangent squared, which is just tangent. We'll be multiplying sin theta by tangent of data in the denominator, and then that gets multiplied by the data. The tangent of the will cancel out, so this is just gonna be equal to the integral of one divided by. Deacon of data. The data. However, one divided by C and theta is just cosine theta. So this is equal to the interval of cosine theta. The data and perform this integration, when we integrate cosine, this is going to be equal to kind of data. Plus integration constant, so we'll have plus C. Now we just need to rewrite this in terms of X. So coming back to our definition. We see that theta. Using our definition of X is going to be equal to. In verse beacon Of the quantity of X in quantity. So this means that sine of theta is going to be equal to the square root of the quantity of the quantity of 2 X in quantity squared minus 1 in quantity divided by the quantity of X. So, coming back, our original interval of 1 divided by the quantity of X2 multiplied by the square root of the quantity of 16 X2 minus 4 in the quantity DX. is going to be equal to. The square root of the quantity of 4 x 2 minus 1 in quantity, divided by the quantity of 2 X in quantity plus D. So this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
42. ∫ 1/(x²√(9x² - 1)) dx, x > 1/3

Key Concepts

Trigonometric Substitution

Integration Techniques

Definite vs. Indefinite Integrals

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