Please examine the texts provided thoroughly to identify and correct any spelling, typographical, grammatical, OCR (optical character recognition), and mathematical errors, including any errors related to the square root symbol. Ensure that the entire text is properly formatted and presented in a clear, coherent manner. Only correct errors, Do not answer it.
####
Texts: √√81-²2² X 9 Video Example) sin(0) = EXAMPLE 1 Evaluate 81-x² SOLUTION Let x = 9 sin(0), where -π/2 ≤ 0 ≤ π/2. Then dx = cot(0) = +2 dx = dx. √81-x² = √(81 - 81 sin²(0)) = √(81 cos² (0)) = 9|cos(0)| = 9 cos(0). (Note that cos(0) ≥ 0 because -π/2 ≤ 0 ≤ π/2.) Thus the Inverse Substitution Rule gives | √(81-x²) +² X 9 cos(0) - √(31 Sin ² (0)) 81 cos² (0) - [ cot² (0) de = = [(csc²(0) - 1) de de + C. de and x) de calcPa Operatio Function: Since this is an indefinite integral, we must return to the original variable x. This can be done either by using trigonometric identities to express cot(0) in terms of sin(0) = x/9 or by drawing a diagram, as in the Figure, where 0 is interpreted as an angle of a right triangle. Since sin(0) = x/9, we label the opposite side and the hypotenuse as having lengths x and 9. Then the Pythagorean Theorem gives the length of the adjacent side as √(81-x²), so we can simply read the value of cot(0) from the figure. (Although > 0 in the diagram, this expression for cot(0) is valid even when 0 < 0.) Since sin(0) = x/9, we have 0 = sin⁻¹(x/9) and so on.
