Evaluate the integrals in Exercises 31–78.
73. ∫dx/√(-2x...

Question

Evaluate the integrals in Exercises 31–78.

73. ∫dx/√(-2x-x²)

Accepted Answer

Welcome back everyone. Evaluate the integral dx divided by the square root of -4 x minus x2. Let's go ahead and rewrite the integral, integral of dx divided by the square root of -4 x minus x2. And let's begin this problem by completing the square. What we can do is rewrite the integral as dx divided by the square root of. Let's go ahead and factor out the negative sign, so that being negative in parent x2 + 4 x. From here, we're going to complete the square, so that'd be the integral of d x divided by. Square root of Negative in parentheses in order to complete the square, we are simply going to add 4, right? So that would be negative in parent x2 + 4 x + 4. And now let's think about it. If we distribute the negative sign, we're going to have 4, so we must also add 4, right, to ensure that the original expression is unchanged. And now let's go ahead and rewrite the integral as d x divided by. Square root Of We can begin with the positive terms we have 4 minus and notice that we have a perfect square which is x plus. 22. So we take 4 minus x plus 22. So this is how we complete the square, and now we can use the u substitution. Let's suppose that u is equal to x + 2, and let's remember that the next step is to find the derivative of u with respect to x. That's the derivative of x + 2, which is equal to 1, and it essentially means that d u is equal to dx because the ratio is 1. So we can rewrite the integral as the integral of DU. Divided by the square root of 4 minus u 2. Now this is one of the elementary integrals. If we rewrite for s 22, notice that it corresponds to the formula integral of d u divided by square root of a 2 minus u 2. And according to the formula, the value of this integral is going to be inverse of sin of u divided by a. Notice that in the denominator we have the difference between a 2 and u2. So if a2 is 22, a is equal to 2, right? So we're going to get inverse of sin of u divided by 2 plus a constant of integration c. And now We can replace you with X + 2 to get our final answer. That's inverse of sign of x + 2 divided by 2 + C. Let's label it and thank you for watching.

Evaluate the integrals in Exercises 31–78.73. ∫dx/√(-2x-x²)

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Evaluate the integrals in Exercises 31–78.
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