Evaluate the integrals in Exercises 77–90.
90. ∫dx/((x-2...

Question

Evaluate the integrals in Exercises 77–90.

90. ∫dx/((x-2)√(x²-4x+3))

Accepted Answer

Welcome back everyone. Evaluate the integral dx divided by x + 1 multiplied by the square root of x2 + 2 x minus 3. For this problem we're going to make use of the integral of d u divided by u which is multiplied by a square root of. U2 minus a 2. Remember that the value of this integral is 1 divided by a multiplied by inverse of 2 of u divided by a plus c. So to use this formula we essentially want to first of all complete the square of the quadratic polynomial. We have x2 + 2 x minus 3, and we can basically write it as x + 12 minus 4, right? We get x2 + 2 x + 1, and in order to get -3, we have to subtract 4. So now we can rewrite our integral as. Integral of d x divided by x plus 1 square root of X + 12 minus 4 and notice that we can introduce a substitution u equals x + 1. The next step is to find the derivative of u with respect to x. That's the derivative of x plus 1, which is equal to 1, and it basically implies that duu is equal to dx. So the integral can be written as the integral of d u because dx becomes the u. Divided by u square root of U 2 minus 4 we can now rewrite 4s22 right to clearly see that a is 2 and apply the formula. This is going to be 1 divided by A, which is 1/2. Inverse of 2 of u divided by 2 again a is 2 plus a constant of integration c. And finally we're going to replace you with x + 1. To get 1/2. Inverse of 2 of x + 1 divided by 2 + c. That's our final answer, and thank you for watching.

Evaluate the integrals in Exercises 77–90.90. ∫dx/((x-2)√(x²-4x+3))

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Evaluate the integrals in Exercises 77–90.
90. ∫dx/((x-2)√(x²-4x+3))