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Integrate 19−x2\frac{1}{\sqrt{9-x^2}}

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Full solution

Q. Integrate 19−x2\frac{1}{\sqrt{9-x^2}}
  1. Identify integral: Identify the integral to solve.\newlineIntegrate 19−x2 dx\frac{1}{\sqrt{9-x^2}} \, dx.
  2. Recognize standard form: Recognize the integral as a standard form.\newlineThis is the integral of the form ∫1a2−x2 dx\int \frac{1}{\sqrt{a^2 - x^2}} \, dx, which is arcsin⁡(xa)+C\arcsin(\frac{x}{a}) + C.
  3. Substitute a value: Substitute a=3a = 3 into the formula.\newline∫19−x2 dx=arcsin⁡(x3)+C\int \frac{1}{\sqrt{9-x^2}} \, dx = \arcsin\left(\frac{x}{3}\right) + C.

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