Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-...

Question

Evaluate the integrals in Exercises 53–76.

69. ∫dx/((2x-1)√((2x-1)²-4))

Accepted Answer

Welcome back everyone. Evaluate the integral dx divided by 3 x + 4 multiplied by square root of 3 x + 42 minus 2 multiplied by 3 x + 4 + 14 x greater than -1. For this problem, we're going to use use substitution. Let's suppose that U is equal to 3 x + 4. Now the derivative of u with respect to x is going to be equal to the derivative of 3 x plus 4, which is 3. So we can now show that dx is equal to duu divided by 3, or basically dx equals 1/3 d u, right? Now we're going to rewrite our integral as follows. So we have integral of dx becomes 1/3. DU. We can also factor out 1/3, and in the denominator we are going to have u. Square root of U 2 minus 2 u + 1. Notice that we have a perfect square, so we're going to get 1/3 integral d u divided by u square root of. Let's write the fact root form that'll be u minus 1 squared. This is going to be equal to 1/3 integral the u divided by. You multiplied by the absolute value of u minus 1. Now what we can do is drop the absolute value, and the reason why we can do that is basically because X is greater than -1. If x is greater than -1, u is basically greater than 3 multiplied by -1 plus 4, right? So u is greater than 1, and therefore we can drop the absolute value because u values are positive. So we get 1/3 integral d u divided by. You multiplied by u minus 1. And from here we can use partial fraction decomposition. Let's rewrite our fraction 1 divided by u. U minus 1 is a divided by the first factor which is u plus b divided by the second factor which is u minus 1. Now, if we find the common denominator, that would be a multiplied by u minus 1. Plus b multiplied by u divided by the common denominator u u minus 1. And if we expand the numerator, that would be Au minus A. Plus b u which is equal to 1. Now, let's go ahead and perform factorization. We are going to have A + B. Multiplied by u minus a equals 1. And from here we can have a system of equations. So first of all, A plus B is going to be 0. That's because we don't have any u terms on the right hand side. And nea is going to be equal to the free coefficient, that's 1. So we can now show that a is going to be equal to -1 from the second equation. And B is going to be equal to A, which is 1. Well done, so now we can rewrite. Our integral, that'll be 1/3 integral. Of We know that our expression becomes a divided by you, which is. -1 divided by you. Plus B divided by u minus 1. So plus 1 divided by u minus 1 followed by DU. You know these are elementary functions, right? So we're going to get 1/3 in parentheses, the integral of negative d u divided by u is negative lm of the absolute value of u. Again, no necessity to add the absolute value because u is greater than 1. And for the second part, we have DU divided by u minus 1. We're going to have ln of u minus 1, right? Essentially, we still have a linear function in the denominator with a slope of 1. So the same rule applies. We're going to have plus ln of u minus 1. Plus a constant of integration c. And now our final step is to use the properties of logarithms and simplify, right? So we're going to have 1/3 LN off. U minus 1 divided by You Plus C. And now let's replace you with the actual expression in terms of X. Let's remember that U was equal to 3 x plus 4, right? So we're going to have 1/3 LN of 3 x + 4 minus 1, so that's 3 x plus 3. Divided by u, which is 3 x plus 4. Plus a constant of integration c. And that's our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 53–76.69. ∫dx/((2x-1)√((2x-1)²-4))

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Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))