Evaluate the integrals in Exercises 67–74 in terms of
b....

Question

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral. The integral evaluated from 16 to 1/4 of d x divided by x multiplied by the square root of 1 minus 4 x 2. OK. So it appears for this particular problem, we're asked to evaluate this integral that is provided to us for the prom itself. So now that we know that we're ultimately trying to evaluate this integral, we need to be able to look at this integral, and we need to recognize that we can use the substitution method. So based on our prior knowledge, we need to recall that we can use a trigometric substitution. So with that in mind, so that's our first step is we need to use trigometric substitution. So, focusing on this particular problem, let us state that 2 X will be equal to sin of theta. So, that will mean that DX by Dheta will be equal to 1/2 multiplied by cosine of theta. Which will mean that DX will be equal to 1/2 multiplied by cosine theta Dheta. Fantastic. So that means that the radical will simplify to. The square root of 1 minus 2 x to the power of 2, which will be equal to the square root of 1 minus sin to the power of 2 of theta, which will be equal to simply cosine of theta. So when we substitute into our integral, we will be able to write that the integral of 1/2 multiplied by cosine of theta d theta divided by parentheses of 1/2 multiplied by sine of theta multiplied by cosine of theta will be equal to the integral of cosecant of theta d theta. Which will be equal to negative, the natural log of the absolute value of cosi canta theta plus cotangent of theta. Plus C or C denotes some constant variable. So now, at this point, we need to convert back to X. So we need to take Cosikta theta is equal to 1 divided by sine of theta, which will be equal to 1 divided by 2 X. And cotangent of theta is equal to cosine theta divided by sine of theta, which will be equal to the square root of 1 minus sin square of theta divided by sine theta. Which will be equal to the square root of 1 minus 4 multiplied by x to the 2 divided by 2 x. Thus, we can go ahead and write that the negative natural log of. Cosicant of theta. So once again, negative natural log of the absolute value of cosicant of theta plus cotangent. Of theta Plus C So once again, plus C, this will be equal to negative the natural log of the absolute value of 1 divided by 2 x plus the square root of 1 minus 4 multiplied by x to the 2 divided by 2 x plus C. And this will be equal to negative, the natural log of the absolute value of 1 plus the square root of 1 minus 4 multiplied by x 2. All divided by, so all divided by. 2 X. Plus C, OK. So our second step now is we need to evaluate our limit. So we need to take the integral evaluated from 16 to 1/4 of DX divided by x multiplied by the square root of 1 minus 4 x. The power 2, which will be equal to negative natural log of the absolute value of 1 plus the square root of 1 minus 4 x 2, all divided by 2 x. In absolute value signs, and we need to evaluate this from 1/6 to 1/4. And this will be equal to negative natural log of the absolute value of 1 plus the square root of 1 minus 4 multiplied by parentheses of 1/4 of 2, all divided by 2 multiplied by 1/4. Plus the natural log of the absolute value of 1 plus the square root of 1 +, or it's gonna be 1 minus 1 minus 4 multiplied by parentheses of 1/6 of 2, all divided by 2 multiplied by 1/6. Fantastic. So, skipping a few steps because ultimately we're trying to find our final answer in the most simplified expression, and when we do that, skipping a few steps, we will find that this will be simply equal to negative, the natural log of the absolute value of. 2 Plus the square root of 3, so 2 plus the square root of 3. Don't forget your absolute value signs, plus the natural log of the absolute value of 3 plus 2 multiplied by the square root of 2. Once again, don't forget your absolute value signs, fantastic. So, moving right along here. That will mean that this will be equal to the natural log of the absolute value of 3 plus, so it's gonna be the natural log of the absolute value of 3 + 2 multiplied by the square root of 2 minus the natural log of the absolute value of 2 plus the square root of 3. Which will be equal to the natural log of the absolute value of 3 + 2 multiplied by the square root of 2, divided by 2 plus the square root. Of 3, which will mean that our final answer will be equal to the natural log of parentheses of 3 + 2 multiplied by the square root of 2 divided by 2 plus the square root of 3, and this will be our final answer. Hooray, we did it. So, you don't have to write your final answer for the natural log inside of the absolute value signs because it will just yield a positive result, and because of that, we can ignore the absolute value signs. So, with that said, Let us go back up to the top to recap what we've learned. So based on what we've learned together, we could safely say that our evaluated integral, once again is the natural log of parentheses 3 + 2 multiplied by the square root of 2, all divided by 2 plus the square root of 3. And that's it, that is our final answer. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the integrals in Exercises 67–74 in terms ofb. natural logarithms.71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

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Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))