Evaluate the integrals in Exercises 53–76.
53. ∫dx/√(9-x...

Question

Evaluate the integrals in Exercises 53–76.

53. ∫dx/√(9-x²)

Accepted Answer

Welcome back everyone. Evaluate the integral of d x divided by square root of 16 minus x 2. A says inverse of sin of x divided by 4 + c. B 4 inverse of sin of x divided by 4 + c. C inverse of cosine of x divided by 4 + c, and D inverse of sin of 4 x plus c. For this problem, let's remember an elementary integral, the x divided by square root of 1 minus x 2. It is equal to inverse of sin of x plus e constant of integration c. So we can manipulate the given integral as follows. We're going to write integral dx divided by the square root of, and we can factor out 16, right? So if we factor out 16, we get 16 in parentheses, 1 minus x 2 divided by 16. Now, let's go ahead and factor out 16 from the radiant. And square root of 16 is equal to 4, so we get integral of d x divided by 4 square root of. 1 minus and now what we can do additionally is rewrite x2 divided by 16s x divided by 42. Let's go ahead and factor out 1/4 from the integral. That'd be 1/4 integral. D x divided by square root of 1 minus. X divided by 4 squared and from here we're going to use the u substitution. Let's suppose that u is equal to x divided by 4. Let's evaluate the derivative with respect to x. The derivative of 1/4 x is equal to 1/4, and we can now show that dx is equal to DU multiplied by 4, right? So DX is equal to 4DU. So now we can continue and show that our integral is 1/4 integral. Dx becomes 4 DU, and we're dividing by square root of. 1 minus u 2 Notice that we can cancel out 4. Because 4 divided by 4 is 1, and we're basically integrating DU divided by the square root of 1 minus u 2, which is an elementary integral. Its value is inverse of sine of u plus c. And now let's replace you. With x divided by 4, so we get inverse of sin of x divided by 4 plus a constant of integration c. And this corresponds to the answer of choice a. Thank you for watching.

Evaluate the integrals in Exercises 53–76.53. ∫dx/√(9-x²)

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Evaluate the integrals in Exercises 53–76.
53. ∫dx/√(9-x²)